Take-home Exercise 2: Applied Spatial Interaction Models: A case study of Singapore public bus commuter flows

Published

December 9, 2023

Modified

January 23, 2024

1 Background

Urban commuters on the morning and traveling to work face complex mobility challenges, prompting questions about the impacts of public bus service removal along their routes. Traditional methods like commuter surveys are costly, time-consuming, and often result in outdated information. With the digitization of city-wide infrastructure and the widespread use of GPS on the vehicles and SMART cards in public transport, vast geospatial data sets are generated. However, the rapid growth of this data has overwhelmed planners, hindering their ability to transform it into meaningful insights and impacting the return on investment in data collection.

2 The Data

2.1 Open Government Data

For the purpose of this assignment, data from several open government sources will be used:

  • Passenger Volume by Origin Destination Bus Stops, Bus Stop Location, Train Station and Train Station Exit Point, just to name a few of them, from LTA DataMall.

  • Master Plan 2019 Subzone Boundary, HDB Property Information, School Directory and Information and other relevant data from Data.gov.sg.

  • hexagon, a hexagon layer of 375m (this distance is the perpendicular distance between the centre of the hexagon and its edges.) should be used to replace the relative coarse and irregular Master Plan 2019 Planning Sub-zone GIS data set of URA.

2.2 Specially collected data

  • Business, entertn, F&B, Leisure&Recreation are geospatial data sets of the locations of business establishments, entertainments, food and beverage outlets, leisure and recreation centres compiled for case study of Singapore public bus commuter flows. They are available on in the geospatial folder to Take-home Exercise 2 data folder.

  • HDB: This data set is the geocoded version of HDB Property Information data from data.gov. The data set is prepared using September 2021 data.

2 Getting Started

For the purpose of this study, 10 r packages will be used. They are:

  1. tmap is a package for creating thematic maps. It provides a flexible and powerful framework for visualizing spatial data.

  2. sf is an R package for simple features, which provides a standard way to encode spatial vector data.

  3. DT is a package that extends the data.table package and provides functions to create interactive data tables in R Markdown documents and Shiny web applications.

  4. stplanr is a package for sustainable transport planning. It offers functions for transport planning and modeling, particularly with spatial data.

  5. performance package is often associated with the yardstick package and provides functions for model performance metrics, such as RMSE, MAE, etc.

  6. sp is a package for handling and analyzing spatial data. It is one of the foundational packages for spatial data manipulation and analysis in R.

  7. ggpubr is a package for creating beautiful and customizable ggplot2-based plots for publication.

  8. tidyverse is a collection of R packages, including dplyr, ggplot2, tidyr, and others. It promotes a tidy data workflow and provides a consistent set of functions for data manipulation and visualization.

  9. corrplot is a package for visualizing correlation matrices using colored plots. It helps in understanding the relationships between variables in a dataset.

  10. yardstick is a package for measuring model performance. It provides functions for computing various metrics, such as accuracy, precision, recall, and others.

The code chunk below installs and loads the various packages.

pacman::p_load(tmap, sf, DT, stplanr,   
               performance, sp,
               ggpubr, tidyverse, corrplot,
               yardstick)

3 Data Preparation

3.1 Importing the OD data

Firstly, we will import the Passenger Volume by Origin Destination Bus Stops data set downloaded from LTA DataMall by using read_csv() of readr package.

Code
odbus <- read_csv("data/aspatial/origin_destination_bus_202308.csv",show_col_types = FALSE)

Let use display the odbus tibble data table by using the code chunk below.

glimpse(odbus)
Rows: 5,709,512
Columns: 7
$ YEAR_MONTH          <chr> "2023-08", "2023-08", "2023-08", "2023-08", "2023-…
$ DAY_TYPE            <chr> "WEEKDAY", "WEEKENDS/HOLIDAY", "WEEKENDS/HOLIDAY",…
$ TIME_PER_HOUR       <dbl> 16, 16, 14, 14, 17, 17, 17, 17, 7, 17, 14, 10, 10,…
$ PT_TYPE             <chr> "BUS", "BUS", "BUS", "BUS", "BUS", "BUS", "BUS", "…
$ ORIGIN_PT_CODE      <chr> "04168", "04168", "80119", "80119", "44069", "4406…
$ DESTINATION_PT_CODE <chr> "10051", "10051", "90079", "90079", "17229", "1722…
$ TOTAL_TRIPS         <dbl> 7, 2, 3, 10, 5, 4, 3, 22, 3, 3, 7, 1, 3, 1, 3, 1, …

A quick check of odbus tibble data frame shows that the values in OROGIN_PT_CODE and DESTINATON_PT_CODE are in numeric data type. Hence, the code chunk below is used to convert these data values into character data type.

odbus$ORIGIN_PT_CODE <- as.factor(odbus$ORIGIN_PT_CODE)
odbus$DESTINATION_PT_CODE <- as.factor(odbus$DESTINATION_PT_CODE) 

3.2 Extracting the study data

For this take home exercise, we will extract commuting flows on weekday and between 6 and 9 o’clock.

odbus6_9 <- odbus %>%
  filter(DAY_TYPE == "WEEKDAY") %>%
  filter(TIME_PER_HOUR >= 6 &
           TIME_PER_HOUR <= 9) %>%
  group_by(ORIGIN_PT_CODE,
           DESTINATION_PT_CODE) %>%
  summarise(TRIPS = sum(TOTAL_TRIPS))

Table below shows the content of odbus6_9

Code
datatable(odbus6_9)

4 Working with Geospatial Data

For the purpose of this exercise, three geospatial data will be used. They are:

  • Bus Stop Location from LTA DataMall. It provides information about all the bus stops currently being serviced by buses, including the bus stop code (identifier) and location coordinates.

  • MPSZ-2019: This data provides the sub-zone boundary of URA Master Plan 2019.

  • hexagon, a hexagon layer of 375m (this distance is the perpendicular distance between the centre of the hexagon and its edges.) should be used to replace the relative coarse and irregular Master Plan 2019 Planning Sub-zone GIS data set of URA.

4.1 Importing geospatial data

busstop <- st_read(dsn = "data/geospatial",
                   layer = "BusStop") %>%
  st_transform(crs = 3414)
Reading layer `BusStop' from data source 
  `C:\cjh202311\isss624\Take-home_Ex\Take-home_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 5161 features and 3 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 3970.122 ymin: 26482.1 xmax: 48284.56 ymax: 52983.82
Projected CRS: SVY21

Let’s take a look at the location of bus stops in Singapore:

plot(busstop['BUS_STOP_N'])

mpsz <- st_read(dsn = "data/geospatial",
                   layer = "MPSZ-2019") %>%
  st_transform(crs = 3414)
Reading layer `MPSZ-2019' from data source 
  `C:\cjh202311\isss624\Take-home_Ex\Take-home_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 332 features and 6 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 103.6057 ymin: 1.158699 xmax: 104.0885 ymax: 1.470775
Geodetic CRS:  WGS 84

Let’s take a look at the map of subzone in Singapore.

Code
plot(mpsz['SUBZONE_C'])

4.2 Create Hexagon grid (honeycomb)

For this exercise purpose, we need to create an analytical hexagon data of 375m. (this distance is the perpendicular distance between the centre of the hexagon and its edges)

The following code creates a hexagon grid, associates bus stops with hexagons, and filters out hexagons without any bus stops. The resulting bs_num data frame contains information about hexagons with at least one bus stop.

area_honeycomb_grid = st_make_grid(busstop,cellsize = 750, what = "polygons", square = FALSE)

honeycomb_grid_sf = st_sf(area_honeycomb_grid) %>%
  mutate(grid_id = 1:length(lengths(area_honeycomb_grid)))
honeycomb_grid_sf$num_bs = lengths(st_intersects(honeycomb_grid_sf, busstop))


bs_num = filter(honeycomb_grid_sf, num_bs > 0)

Let’s take a look at the hexagon grid:

Code
tmap_mode("plot")
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz) +
  tm_borders(alpha=0.7,col = "white")+
  tm_polygons("darkgrey") +
tm_shape(bs_num) +
  tm_fill("num_bs",alpha=0.8,palette = "Greens")+
  tm_polygons()

5 Geospatial data wrangling

5.1 Combining Busstop and Hexagon grid

After running this code, honeycomb_grid will be a data frame containing information about bus stops that fall within the hexagons, including the bus stop ID (BUS_STOP_N), the hexagon ID (grid_id), and the number of bus stops within that hexagon (num_bs).

honeycomb_grid <- st_intersection(busstop, bs_num) %>%
  select(BUS_STOP_N, grid_id,num_bs) %>%
  st_drop_geometry()

5.2 Combine odbus6_9 with od_data

For further visual operation, we need to combine a table contains the number of trips between pairs of origin and destination hexagons.

od_data <- left_join(odbus6_9, honeycomb_grid,
                      by = c("ORIGIN_PT_CODE" = "BUS_STOP_N")) %>%
  rename(ORIGIN_BS = ORIGIN_PT_CODE,
         ORIGIN_SZ = grid_id,
         DESTIN_BS = DESTINATION_PT_CODE)

Before continue, check for duplicating records.

duplicate <- od_data %>%
  group_by_all() %>%
  filter(n()>1) %>%
  ungroup()

Another left join between odbus6_9 and honeycomb_grid to get the DESTIN_BS.

od_data <- left_join(od_data, honeycomb_grid, 
                           by = c("DESTIN_BS" = "BUS_STOP_N")) %>%
  rename(DESTIN_SZ = grid_id)

Before continue, check for duplicating records.

duplicate <- od_data %>%
  group_by_all() %>%
  filter(n()>1) %>%
  ungroup()

Group by ‘ORIGIN_GRID_ID’ and ‘DESTIN_GRID_ID’ to create a new column ‘MORNING_PEAK’ representing the total number of trips between each pair of hexagons.

od_data <- od_data %>%
  drop_na() %>%
  group_by(ORIGIN_SZ, DESTIN_SZ) %>%
  summarise(MORNING_PEAK = sum(TRIPS))

6 Visualising Spatial Interaction

6.1 Removing intra-zonal flows

We will not plot the intra-zonal flows. The code chunk below will be used to remove intra-zonal flows.

od_data1 <- od_data[od_data$ORIGIN_SZ!=od_data$DESTIN_SZ,]

6.2 Create flowline

flowLine <- od2line(flow = od_data1, 
                    zones = bs_num,
                    zone_code = "grid_id")

6.3 Visualisation and Analysis

Code
tmap_mode("plot")
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz) +
  tm_borders(alpha=0.7,col = "white")+
  tm_polygons("darkgrey") +
tm_shape(bs_num) +
  tm_fill("num_bs",alpha=0.8,palette = "Greens")+
  tm_polygons() +
flowLine %>%  
  filter(MORNING_PEAK >= 5000) %>%
tm_shape() +
  tm_lines(lwd = "MORNING_PEAK",
           style = "quantile",
           scale = c(0.3,3, 6, 9, 12, 18,21, 27),
           n = 6,
           alpha = 0.5,
           col="yellow") +
  tm_layout(main.title = 'OD Flow On Weekday Morning Peak hour' ,
            main.title.position = "center",
            main.title.size = 1,
            main.title.fontface = 'bold') +
  tm_compass(type="8star", size = 1) +
  tm_scale_bar()

In this map, we filtered out trips with a count less than 5000 for enhanced analysis. Thicker flowlines indicate a higher volume of trips, while the length of the flowlines represents the distance between each hexagon. Notably, we observe long lines from the northern to the eastern regions, which, upon referencing the Singapore subway map, suggests a lack of subway connections along that route. Additionally, thicker lines predominantly appear in darker-colored hexagons, indicating a positive correlation between the number of bus stations and trip volume.

Code
tmap_mode("plot")
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz) +
  tm_borders(alpha=0.7,col = "white")+
  tm_polygons("darkgrey") +

tm_shape(bs_num %>%
  filter(num_bs <= 3)) +
  tm_fill("num_bs",alpha=0.8,palette = "Greens")+
  tm_polygons() +

tm_shape(flowLine %>%  
  filter(MORNING_PEAK >= 20000)) +
  tm_lines(lwd = "MORNING_PEAK",
           style = "quantile",
           scale = c(0.3,3, 6, 9, 12, 18,21, 27),
           n = 6,
           alpha = 0.5,
           col="yellow") +
  tm_layout(main.title = 'OD Flow On Weekday Morning Peak hour' ,
            main.title.position = "center",
            main.title.size = 1,
            main.title.fontface = 'bold') +
  tm_compass(type="8star", size = 1) +
  tm_scale_bar()

After filtering for bus stops with a count of 3 or fewer and trips exceeding 20000, we observe a significant reduction in the overlap between flow lines and hexagons. This confirms the earlier understanding of the relationship between bus stations and trip volume. However, notable overlaps persist in the northern part of Singapore. Given the geographical context, this is likely attributed to buses traveling between Malaysia and Singapore—routes with fewer stations but higher trip volumes.

7 Assemble propulsive and attractiveness variables

business <- st_read(dsn = "data/geospatial",
                   layer = "Business") %>%
  st_transform(crs = 3414)
Reading layer `Business' from data source 
  `C:\cjh202311\isss624\Take-home_Ex\Take-home_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 6550 features and 3 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 3669.148 ymin: 25408.41 xmax: 47034.83 ymax: 50148.54
Projected CRS: SVY21 / Singapore TM
bs_num$`BUSINESS_num`<- lengths(
  st_intersects(
    bs_num, business))
summary(bs_num$BUSINESS_num)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   0.000   1.000   7.273   7.000  97.000 
Code
tmap_mode("plot")
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz) +
  tm_borders(alpha=0.7,col = "white")+
  tm_polygons("darkgrey") +

tm_shape(bs_num) +
  tm_polygons(alpha=0.5, col = "darkgreen")+
tm_shape(business) +
  tm_dots(size = 0.05, col = "yellow") 

fb <- st_read(dsn = "data/geospatial",                    layer = "F&B") %>%   st_transform(crs = 3414)
Reading layer `F&B' from data source 
  `C:\cjh202311\isss624\Take-home_Ex\Take-home_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 1919 features and 3 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 6010.495 ymin: 25343.27 xmax: 45462.43 ymax: 48796.21
Projected CRS: SVY21 / Singapore TM
bs_num$`FB_num`<- lengths(   
  st_intersects(     
    bs_num, fb))
summary(bs_num$FB_num)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   0.000   0.000   2.204   1.000 133.000 
Code
tmap_mode("plot") 
tmap_options(check.and.fix = TRUE) 
tm_shape(mpsz) +   
  tm_borders(alpha=0.7,col = "white")+   
  tm_polygons("darkgrey") +  tm_shape(bs_num) +   
  tm_polygons(alpha=0.5, col = "darkgreen")+ 
  tm_shape(fb) +   
  tm_dots(size = 0.05, col = "yellow") 

lr <- st_read(dsn = "data/geospatial",                    
              layer = "Liesure&Recreation") %>%   
  st_transform(crs = 3414)
Reading layer `Liesure&Recreation' from data source 
  `C:\cjh202311\isss624\Take-home_Ex\Take-home_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 1217 features and 30 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 6010.495 ymin: 25134.28 xmax: 48439.77 ymax: 50078.88
Projected CRS: SVY21 / Singapore TM
bs_num$`lr_num`<- lengths(   
  st_intersects(     
    bs_num, lr))
summary(bs_num$lr_num)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   0.000   0.000   1.307   1.000  41.000 
Code
tmap_mode("plot") 
tmap_options(check.and.fix = TRUE) 
tm_shape(mpsz) +   
  tm_borders(alpha=0.7,col = "white")+   
  tm_polygons("darkgrey") +  tm_shape(bs_num) +   
  tm_polygons(alpha=0.5, col = "darkgreen")+ 
  tm_shape(lr) +   
  tm_dots(size = 0.05, col = "yellow") 

schools <- read_csv("data/aspatial/schools.csv") %>%
  rename(latitude = "results.LATITUDE",
         longitude = "results.LONGITUDE")%>%
  select(postal_code, school_name, latitude, longitude)
schools_sf <- st_as_sf(schools, 
                       coords = c("longitude", "latitude"),
                       crs=4326) %>%
  st_transform(crs = 3414)
bs_num$`schools_num`<- lengths(   
  st_intersects(     
    bs_num, schools_sf))
summary(bs_num$schools_num)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.0000  0.0000  0.4125  1.0000  4.0000 
Code
tmap_mode("plot") 
tmap_options(check.and.fix = TRUE) 
tm_shape(mpsz) +   
  tm_borders(alpha=0.7,col = "white")+   
  tm_polygons("darkgrey") +  tm_shape(bs_num) +   
  tm_polygons(alpha=0.5, col = "darkgreen")+ 
  tm_shape(schools_sf) +   
  tm_dots(size = 0.05, col = "yellow") 

hdb <- read_csv("data/aspatial/hdb.csv") %>%
  rename(latitude = "lat",
         longitude = "lng")
hdb_sf <- st_as_sf(hdb, 
                       coords = c("longitude", "latitude"),
                       crs=4326) %>%
  st_transform(crs = 3414)
bs_num$`hdb_num`<- lengths(   
  st_intersects(     
    bs_num, hdb_sf))
summary(bs_num$hdb_num)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   0.00    0.00    0.00   14.85   27.00  103.00 
Code
tmap_mode("plot") 
tmap_options(check.and.fix = TRUE) 
tm_shape(mpsz) +   
  tm_borders(alpha=0.7,col = "white")+   
  tm_polygons("darkgrey") +  tm_shape(bs_num) +   
  tm_polygons(alpha=0.5, col = "darkgreen")+ 
  tm_shape(hdb_sf) +   
  tm_dots(size = 0.05, col = "yellow") 

entertn <- st_read(dsn = "data/geospatial",
                   layer = "entertn") %>%
  st_transform(crs = 3414)
Reading layer `entertn' from data source 
  `C:\cjh202311\isss624\Take-home_Ex\Take-home_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 114 features and 3 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 10809.34 ymin: 26528.63 xmax: 41600.62 ymax: 46375.77
Projected CRS: SVY21 / Singapore TM
bs_num$`entertn_num`<- lengths(
  st_intersects(
    bs_num, entertn))
summary(bs_num$entertn_num)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.0000  0.0000  0.1331  0.0000  9.0000 
Code
tmap_mode("plot")
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz) +
  tm_borders(alpha=0.7,col = "white")+
  tm_polygons("darkgrey") +

tm_shape(bs_num) +
  tm_polygons(alpha=0.5, col = "darkgreen")+
tm_shape(entertn) +
  tm_dots(size = 0.05, col = "yellow") 

The data will be joined with :

bs_num_tidy <- bs_num %>%
  st_drop_geometry() %>%
  select(grid_id, BUSINESS_num, FB_num, lr_num, schools_num,  hdb_num, entertn_num )
flow_data <- od_data1 %>%
  left_join(bs_num_tidy,
            by = c("DESTIN_SZ" = "grid_id"))
Code
summary(flow_data)
   ORIGIN_SZ      DESTIN_SZ     MORNING_PEAK      BUSINESS_num   
 Min.   :  23   Min.   :  23   Min.   :    1.0   Min.   : 0.000  
 1st Qu.:1099   1st Qu.:1116   1st Qu.:    7.0   1st Qu.: 0.000  
 Median :1433   Median :1432   Median :   38.0   Median : 1.000  
 Mean   :1416   Mean   :1414   Mean   :  385.1   Mean   : 7.302  
 3rd Qu.:1728   3rd Qu.:1711   3rd Qu.:  180.0   3rd Qu.: 6.000  
 Max.   :2505   Max.   :2505   Max.   :96644.0   Max.   :97.000  
     FB_num            lr_num        schools_num        hdb_num      
 Min.   :  0.000   Min.   : 0.000   Min.   :0.0000   Min.   :  0.00  
 1st Qu.:  0.000   1st Qu.: 0.000   1st Qu.:0.0000   1st Qu.:  0.00  
 Median :  0.000   Median : 1.000   Median :0.0000   Median : 10.00  
 Mean   :  5.776   Mean   : 2.368   Mean   :0.5291   Mean   : 19.39  
 3rd Qu.:  2.000   3rd Qu.: 3.000   3rd Qu.:1.0000   3rd Qu.: 34.00  
 Max.   :133.000   Max.   :41.000   Max.   :4.0000   Max.   :103.00  
  entertn_num   
 Min.   :0.000  
 1st Qu.:0.000  
 Median :0.000  
 Mean   :0.336  
 3rd Qu.:0.000  
 Max.   :9.000  

We can see there are 0 values in BUSINESS_num, FB_num, lr_num, schools_num, hdb_num, entertn_num.

The code chunk below will be used to replace zero values to 0.99 for the propulsive and attractiveness variables.

flow_data$BUSINESS_num <- ifelse(
  flow_data$BUSINESS_num == 0,
  0.99, flow_data$BUSINESS_num)
flow_data$FB_num <- ifelse(
  flow_data$FB_num == 0,
  0.99, flow_data$FB_num)
flow_data$lr_num <- ifelse(
  flow_data$lr_num == 0,
  0.99, flow_data$lr_num)
flow_data$schools_num <- ifelse(
  flow_data$schools_num == 0,
  0.99, flow_data$schools_num)
flow_data$hdb_num <- ifelse(
  flow_data$hdb_num == 0,
  0.99, flow_data$hdb_num)
flow_data$entertn_num<- ifelse(
  flow_data$entertn_num == 0,
  0.99, flow_data$entertn_num)
Code
summary(flow_data)
   ORIGIN_SZ      DESTIN_SZ     MORNING_PEAK      BUSINESS_num   
 Min.   :  23   Min.   :  23   Min.   :    1.0   Min.   : 0.990  
 1st Qu.:1099   1st Qu.:1116   1st Qu.:    7.0   1st Qu.: 0.990  
 Median :1433   Median :1432   Median :   38.0   Median : 1.000  
 Mean   :1416   Mean   :1414   Mean   :  385.1   Mean   : 7.729  
 3rd Qu.:1728   3rd Qu.:1711   3rd Qu.:  180.0   3rd Qu.: 6.000  
 Max.   :2505   Max.   :2505   Max.   :96644.0   Max.   :97.000  
     FB_num            lr_num        schools_num       hdb_num      
 Min.   :  0.990   Min.   : 0.990   Min.   :0.990   Min.   :  0.99  
 1st Qu.:  0.990   1st Qu.: 0.990   1st Qu.:0.990   1st Qu.:  0.99  
 Median :  0.990   Median : 1.000   Median :0.990   Median : 10.00  
 Mean   :  6.306   Mean   : 2.769   Mean   :1.171   Mean   : 19.76  
 3rd Qu.:  2.000   3rd Qu.: 3.000   3rd Qu.:1.000   3rd Qu.: 34.00  
 Max.   :133.000   Max.   :41.000   Max.   :4.000   Max.   :103.00  
  entertn_num   
 Min.   :0.990  
 1st Qu.:0.990  
 Median :0.990  
 Mean   :1.183  
 3rd Qu.:0.990  
 Max.   :9.000  

Now, there is no zero values in our data.

Next we will remove duplicate record:

duplicate <- flow_data %>%
  group_by_all() %>%
  filter(n()>1) %>%
  ungroup()
flow_data <- unique(flow_data)
Code
summary(flow_data)
   ORIGIN_SZ      DESTIN_SZ     MORNING_PEAK      BUSINESS_num   
 Min.   :  23   Min.   :  23   Min.   :    1.0   Min.   : 0.990  
 1st Qu.:1099   1st Qu.:1116   1st Qu.:    7.0   1st Qu.: 0.990  
 Median :1433   Median :1432   Median :   38.0   Median : 1.000  
 Mean   :1416   Mean   :1414   Mean   :  385.1   Mean   : 7.729  
 3rd Qu.:1728   3rd Qu.:1711   3rd Qu.:  180.0   3rd Qu.: 6.000  
 Max.   :2505   Max.   :2505   Max.   :96644.0   Max.   :97.000  
     FB_num            lr_num        schools_num       hdb_num      
 Min.   :  0.990   Min.   : 0.990   Min.   :0.990   Min.   :  0.99  
 1st Qu.:  0.990   1st Qu.: 0.990   1st Qu.:0.990   1st Qu.:  0.99  
 Median :  0.990   Median : 1.000   Median :0.990   Median : 10.00  
 Mean   :  6.306   Mean   : 2.769   Mean   :1.171   Mean   : 19.76  
 3rd Qu.:  2.000   3rd Qu.: 3.000   3rd Qu.:1.000   3rd Qu.: 34.00  
 Max.   :133.000   Max.   :41.000   Max.   :4.000   Max.   :103.00  
  entertn_num   
 Min.   :0.990  
 1st Qu.:0.990  
 Median :0.990  
 Mean   :1.183  
 3rd Qu.:0.990  
 Max.   :9.000  

The following table is explanatory variables to be used in the Spatial Interaction Modelling:

No Explanatory Variables
1 Business
2 F&B
3 Leisure&Rrecreation
4 schools
5 HDB
6 Entertainment

8 Compute Distance Matrix

8.1 Converting from sf data.table to SpatialPolygonsDataFrame

First as.Spatial() will be used to convert bs_num from sf tibble data frame to SpatialPolygonsDataFrame of sp object as shown in the code chunk below. Computing distance matrix using sp than sf package.

bs_num_sp <- as(bs_num, "Spatial")

8.2 Computing the distance matrix

Next, spDists() of sp package will be used to compute the Euclidean distance between the centroids of the hexagonsn.

The following code calculates the pairwise distances between the bus stations represented by bs_num_sp. The spDists function from the sp package is employed for this purpose. The distances are computed based on the spatial coordinates of the bus stations, and the result is stored in the variable dist. The parameter longlat = FALSE indicates that the distances should be computed in the native coordinate system (not considering latitude and longitude as spherical coordinates).

dist <- spDists(bs_num_sp, 
                longlat = FALSE)
head(dist, n=c(10, 10))
          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]     [,8]
 [1,]    0.000  750.000 3269.174 1500.000 2704.163 3968.627 1299.038 2250.000
 [2,]  750.000    0.000 2598.076  750.000 1984.313 3269.174  750.000 1500.000
 [3,] 3269.174 2598.076    0.000 1984.313  750.000  750.000 2704.163 1500.000
 [4,] 1500.000  750.000 1984.313    0.000 1299.038 2598.076  750.000  750.000
 [5,] 2704.163 1984.313  750.000 1299.038    0.000 1299.038 1984.313  750.000
 [6,] 3968.627 3269.174  750.000 2598.076 1299.038    0.000 3269.174 1984.313
 [7,] 1299.038  750.000 2704.163  750.000 1984.313 3269.174    0.000 1299.038
 [8,] 2250.000 1500.000 1500.000  750.000  750.000 1984.313 1299.038    0.000
 [9,] 3436.932 2704.163  750.000 1984.313  750.000  750.000 2598.076 1299.038
[10,] 4683.748 3968.627 1500.000 3269.174 1984.313  750.000 3897.114 2598.076
          [,9]    [,10]
 [1,] 3436.932 4683.748
 [2,] 2704.163 3968.627
 [3,]  750.000 1500.000
 [4,] 1984.313 3269.174
 [5,]  750.000 1984.313
 [6,]  750.000  750.000
 [7,] 2598.076 3897.114
 [8,] 1299.038 2598.076
 [9,]    0.000 1299.038
[10,] 1299.038    0.000

8.3 Labelling column and row heanders of a distance matrix

First, we will create a list sorted according to the the distance matrix by grid_id.

grid_id_names <- bs_num$grid_id

Next we will attach grid_id to row and column for distance matrix matching ahead.

colnames(dist) <- paste0(grid_id_names)
rownames(dist) <- paste0(grid_id_names)

8.3 Pivoting distance value by Hexagon grid

We will pivot the distance matrix into a long table by using the row and column grid_id as show in the code chunk below.

The melt operation reshapes the matrix into a long format.

distPair <-  reshape2::melt(dist) %>%
  rename(dist = value)
head(distPair, 10)
   Var1 Var2     dist
1    23   23    0.000
2    44   23  750.000
3    46   23 3269.174
4    66   23 1500.000
5    67   23 2704.163
6    68   23 3968.627
7    86   23 1299.038
8    87   23 2250.000
9    88   23 3436.932
10   89   23 4683.748

8.4 Updating intra-zonal distances

First, we will select and find out the minimum value of the distance by using summary().

distPair %>%
  filter(dist > 0) %>%
  summary()
      Var1           Var2           dist      
 Min.   :  23   Min.   :  23   Min.   :  750  
 1st Qu.: 871   1st Qu.: 871   1st Qu.: 8352  
 Median :1324   Median :1324   Median :13332  
 Mean   :1269   Mean   :1269   Mean   :14162  
 3rd Qu.:1688   3rd Qu.:1688   3rd Qu.:18929  
 Max.   :2505   Max.   :2505   Max.   :44680  

The minimum distance is about 750, indicating travelling to adjacent hexagon.

Next, a constant distance value of 300 (middle to edge is 325) added into intra-zones distance.

distPair$dist <- ifelse(distPair$dist == 0,
                        300, distPair$dist)

The code chunk below will be used to check the result data.frame.

distPair %>%   
  summary()
      Var1           Var2           dist      
 Min.   :  23   Min.   :  23   Min.   :  300  
 1st Qu.: 871   1st Qu.: 871   1st Qu.: 8250  
 Median :1324   Median :1324   Median :13332  
 Mean   :1269   Mean   :1269   Mean   :14145  
 3rd Qu.:1688   3rd Qu.:1688   3rd Qu.:18929  
 Max.   :2505   Max.   :2505   Max.   :44680  

The code chunk below is used to rename the origin and destination fields.

distPair <- distPair %>%
  rename(ORIGIN_GRID_ID = Var1,
         DESTIN_GRID_ID = Var2)

distPair %>% head()
  ORIGIN_GRID_ID DESTIN_GRID_ID     dist
1             23             23  300.000
2             44             23  750.000
3             46             23 3269.174
4             66             23 1500.000
5             67             23 2704.163
6             68             23 3968.627
glimpse(distPair)
Rows: 695,556
Columns: 3
$ ORIGIN_GRID_ID <int> 23, 44, 46, 66, 67, 68, 86, 87, 88, 89, 90, 109, 110, 1…
$ DESTIN_GRID_ID <int> 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23,…
$ dist           <dbl> 300.000, 750.000, 3269.174, 1500.000, 2704.163, 3968.62…

9 Spatial Interaction Modelling

9.1 Preparing Flow Data

In general, we will calibrate separate Spatial Interaction Models for inter- and intra-zonal flows. we will focus our attention on inter-zonal flow. Hence, we need to exclude the intra-zonal flow from flow_data.

First, two new columns called FlowNoIntra and offset will be created by using the code chunk below.

flow_data$FlowNoIntra <- ifelse(
  flow_data$ORIGIN_SZ == flow_data$DESTIN_SZ, 
  0, flow_data$MORNING_PEAK)
flow_data$offset <- ifelse(
  flow_data$ORIGIN_SZ == flow_data$DESTIN_SZ, 
  0.000001, 1)

Now, we need to combine flow data with distance value:

flow_data$ORIGIN_SZ <- as.factor(flow_data$ORIGIN_SZ)
flow_data$DESTIN_SZ <- as.factor(flow_data$DESTIN_SZ)
distPair$ORIGIN_GRID_ID <- as.factor(distPair$ORIGIN_GRID_ID)
distPair$DESTIN_GRID_ID <- as.factor(distPair$DESTIN_GRID_ID)
flow_data1 <- flow_data %>%
  left_join (distPair,
             by = c("ORIGIN_SZ" = "ORIGIN_GRID_ID",
                    "DESTIN_SZ" = "DESTIN_GRID_ID"))

we need to remove duplicate:

duplicate <- flow_data1 %>%
  group_by_all() %>%
  filter(n()>1) %>%
  ungroup()
SIM_data <- unique(flow_data1)

According to the syntax used to derive values in FlowNoIntra field, all intra-zonal flow will be given a value of 0 or else the original flow values will be inserted.

Next, inter-zonal flow will be selected from flow_data and save into a new output data.frame called inter_zonal_flow by using the code chunk below.

inter_zonal_flow <- SIM_data %>%
  filter(FlowNoIntra > 0)
summary(inter_zonal_flow)
   ORIGIN_SZ       DESTIN_SZ      MORNING_PEAK      BUSINESS_num   
 1960   :  298   1474   :  347   Min.   :    1.0   Min.   : 0.990  
 1474   :  292   1452   :  339   1st Qu.:    7.0   1st Qu.: 0.990  
 1477   :  292   1516   :  326   Median :   38.0   Median : 1.000  
 1496   :  278   1960   :  295   Mean   :  385.1   Mean   : 7.729  
 1623   :  277   1477   :  294   3rd Qu.:  180.0   3rd Qu.: 6.000  
 1749   :  273   1494   :  292   Max.   :96644.0   Max.   :97.000  
 (Other):63252   (Other):63069                                     
     FB_num            lr_num        schools_num       hdb_num      
 Min.   :  0.990   Min.   : 0.990   Min.   :0.990   Min.   :  0.99  
 1st Qu.:  0.990   1st Qu.: 0.990   1st Qu.:0.990   1st Qu.:  0.99  
 Median :  0.990   Median : 1.000   Median :0.990   Median : 10.00  
 Mean   :  6.306   Mean   : 2.769   Mean   :1.171   Mean   : 19.76  
 3rd Qu.:  2.000   3rd Qu.: 3.000   3rd Qu.:1.000   3rd Qu.: 34.00  
 Max.   :133.000   Max.   :41.000   Max.   :4.000   Max.   :103.00  
                                                                    
  entertn_num     FlowNoIntra          offset       dist      
 Min.   :0.990   Min.   :    1.0   Min.   :1   Min.   :  750  
 1st Qu.:0.990   1st Qu.:    7.0   1st Qu.:1   1st Qu.: 3000  
 Median :0.990   Median :   38.0   Median :1   Median : 5408  
 Mean   :1.183   Mean   :  385.1   Mean   :1   Mean   : 6334  
 3rd Qu.:0.990   3rd Qu.:  180.0   3rd Qu.:1   3rd Qu.: 8842  
 Max.   :9.000   Max.   :96644.0   Max.   :1   Max.   :24784  
                                                              

9.2 Visualising the dependent variable

Code
ggplot(data = inter_zonal_flow,
       aes(x = MORNING_PEAK)) +
  geom_histogram(fill="darkgreen")+
  theme_minimal()

We can see that all ‘MORNING_PEAK’ values are all positive values and its distribution is highly skewed to the right side.

Next, let us visualise the relation between the dependent variable and one of the key independent variable in Spatial Interaction Model, namely distance.

Code
ggplot(data = inter_zonal_flow,
       aes(x = dist,
           y = MORNING_PEAK)) +
  geom_point(color = "darkgreen") +
  geom_smooth(method = lm, color = "yellow") +
  theme_minimal()

Their relationship hardly resemble linear relationship.

When we plot the scatter plot by using the log transformed version of both variables, we can see that their relationship is more resemble linear relationship.

Code
ggplot(data = inter_zonal_flow,
       aes(x = log(dist),
           y = log(MORNING_PEAK))) +
  geom_point(color = "darkgreen") +
  geom_smooth(method = lm, color = "yellow")+
  theme_minimal()

9.3 Correlation Analysis

Before I proceed to do SIM, I will determine if there are any correlations between the variables. Conducting a correlation analysis is vital as it reveals linear relationships between variables, aiding in the identification of associations. This process helps avoid multicollinearity issues, optimizing regression models by selecting pertinent variables.

inter_zonal_flow$BUSINESS_num <- as.numeric(inter_zonal_flow$BUSINESS_num)

inter_zonal_flow$FB_num<- as.numeric(inter_zonal_flow$FB_num)

inter_zonal_flow$lr_num <- as.numeric(inter_zonal_flow$lr_num)

inter_zonal_flow$schools_num <- as.numeric(inter_zonal_flow$schools_num)

inter_zonal_flow$hdb_num <- as.numeric(inter_zonal_flow$hdb_num)

inter_zonal_flow$entertn_num <- as.numeric(inter_zonal_flow$entertn_num)

inter_zonal_flow$MORNING_PEAK<- as.numeric(inter_zonal_flow$MORNING_PEAK)

Now, we draw the correlation matrix:

Code
vars.cor <- cor(inter_zonal_flow[, 3:9])

corrplot(
  vars.cor,
  method = "ellipse",
  lower = "ellipse",
  upper = "number",
  tl.pos = "lt",
  diag = FALSE, 
  addCoef.col = "black",
  tl.col = "black",
  col = colorRampPalette(c("yellow", "white","lightgreen"))(100) ,number.cex = 0.7
)

The correlation analysis revealed a notable relationship between the variables “F&B” and “Leisure & Recreation” with a correlation coefficient exceeding 80%. This high correlation suggests a strong linear association between the two variables. As a result, it is advisable to retain only one of them in subsequent analyses to avoid issues of multicol-linearity.

Therefore, we choose to remove “Leisure & Recreation” and remain “F&B”. The following table is new explanatory variables to be used in the Spatial Interaction Modelling:

No Explanatory Variables
1 Business
2 F&B
3 schools
4 HDB
5 Entertainment

9.4 Calibrating Spatial Interaction Models

9.4.1 Unconstrained Spatial Interaction Model

The code chunk used to calibrate to model is shown below:

uncSIM <- glm(formula = MORNING_PEAK ~ 
                log(BUSINESS_num) +
                log(FB_num) +
                log(schools_num) +
                log(hdb_num) +
                log(entertn_num)+
                log(dist),
              family = poisson(link = "log"),
              data = inter_zonal_flow,
              na.action = na.exclude)

the calculated R-squared is 0.12527437, indicating that the model accounts for approximately 12.53% of the variability in the observed MORNING_PEAK.

Code
CalcRSquared <- function(observed,estimated){
  r <- cor(observed,estimated)
  R2 <- r^2
  R2
}

CalcRSquared(uncSIM$data$MORNING_PEAK, uncSIM$fitted.values)
[1] 0.1247437
Code
uncSIM$coefficients
      (Intercept) log(BUSINESS_num)       log(FB_num)  log(schools_num) 
      16.67686922        0.05927062        0.03310465        0.21753349 
     log(hdb_num)  log(entertn_num)         log(dist) 
       0.20942559        0.22414936       -1.41229192 
Code
summary(uncSIM)

Call:
glm(formula = MORNING_PEAK ~ log(BUSINESS_num) + log(FB_num) + 
    log(schools_num) + log(hdb_num) + log(entertn_num) + log(dist), 
    family = poisson(link = "log"), data = inter_zonal_flow, 
    na.action = na.exclude)

Coefficients:
                    Estimate Std. Error z value Pr(>|z|)    
(Intercept)       16.6768692  0.0019208  8682.0   <2e-16 ***
log(BUSINESS_num)  0.0592706  0.0001788   331.4   <2e-16 ***
log(FB_num)        0.0331046  0.0002402   137.8   <2e-16 ***
log(schools_num)   0.2175335  0.0006458   336.9   <2e-16 ***
log(hdb_num)       0.2094256  0.0001380  1517.7   <2e-16 ***
log(entertn_num)   0.2241494  0.0007501   298.8   <2e-16 ***
log(dist)         -1.4122919  0.0002495 -5661.3   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 101244825  on 64961  degrees of freedom
Residual deviance:  62792208  on 64955  degrees of freedom
AIC: 63150592

Number of Fisher Scoring iterations: 7

9.4.2 Origin Constrained Model

Code chunk below shows the calibration of the model by using glm() of R and flow_data.

orcSIM <- glm(formula = MORNING_PEAK ~ 
                ORIGIN_SZ +
                log(BUSINESS_num) +
                log(FB_num) +
                
                log(schools_num) +
                log(hdb_num) +
                log(entertn_num)+
                log(dist) - 1,
              family = poisson(link = "log"),
              data = inter_zonal_flow,
              na.action = na.exclude)

the calculated R-squared is 0.2402228, indicating that the model accounts for approximately 24.02% of the variability in the observed MORNING_PEAK.

Code
CalcRSquared(orcSIM$data$MORNING_PEAK, orcSIM$fitted.values)
[1] 0.2402228
Code
orcSIM$coefficients[818:823]
log(BUSINESS_num)       log(FB_num)  log(schools_num)      log(hdb_num) 
        0.1145699         0.1319546         0.1776087         0.1227611 
 log(entertn_num)         log(dist) 
        0.2532727        -1.4416487 
Code
summary(orcSIM)

Call:
glm(formula = MORNING_PEAK ~ ORIGIN_SZ + log(BUSINESS_num) + 
    log(FB_num) + log(schools_num) + log(hdb_num) + log(entertn_num) + 
    log(dist) - 1, family = poisson(link = "log"), data = inter_zonal_flow, 
    na.action = na.exclude)

Coefficients:
                    Estimate Std. Error   z value Pr(>|z|)    
ORIGIN_SZ23       13.6371930  0.1270169   107.365   <2e-16 ***
ORIGIN_SZ44       14.3998717  0.1414392   101.810   <2e-16 ***
ORIGIN_SZ46       12.7403468  0.0924677   137.782   <2e-16 ***
ORIGIN_SZ66       13.5383719  0.1302054   103.977   <2e-16 ***
ORIGIN_SZ67       14.7573273  0.0259747   568.142   <2e-16 ***
ORIGIN_SZ68       12.8086473  0.1104479   115.970   <2e-16 ***
ORIGIN_SZ86       13.6730129  0.0803349   170.200   <2e-16 ***
ORIGIN_SZ87       15.3667022  0.0739574   207.778   <2e-16 ***
ORIGIN_SZ88       15.5861467  0.0189635   821.902   <2e-16 ***
ORIGIN_SZ89       13.5665982  0.0496043   273.497   <2e-16 ***
ORIGIN_SZ90       12.9063595  0.1231060   104.839   <2e-16 ***
ORIGIN_SZ109      13.5231995  0.0894638   151.158   <2e-16 ***
ORIGIN_SZ110      13.7067486  0.0762736   179.705   <2e-16 ***
ORIGIN_SZ111      16.5576703  0.0114502  1446.065   <2e-16 ***
ORIGIN_SZ112      12.8861335  0.0685468   187.990   <2e-16 ***
ORIGIN_SZ128      16.1234129  0.0232791   692.612   <2e-16 ***
ORIGIN_SZ129      13.1831647  0.1767892    74.570   <2e-16 ***
ORIGIN_SZ130      14.1769985  0.0527377   268.821   <2e-16 ***
ORIGIN_SZ131      14.4127053  0.0316185   455.831   <2e-16 ***
ORIGIN_SZ132      13.2849004  0.0532608   249.431   <2e-16 ***
ORIGIN_SZ133      13.5237671  0.0408696   330.900   <2e-16 ***
ORIGIN_SZ134      15.8966693  0.0242145   656.493   <2e-16 ***
ORIGIN_SZ150      16.2978631  0.0198543   820.871   <2e-16 ***
ORIGIN_SZ151      13.3718598  0.0662482   201.845   <2e-16 ***
ORIGIN_SZ152      12.6979398  0.0824984   153.917   <2e-16 ***
ORIGIN_SZ153      14.0984175  0.0420025   335.656   <2e-16 ***
ORIGIN_SZ154      16.6825926  0.0090750  1838.297   <2e-16 ***
ORIGIN_SZ155      13.5966151  0.0355186   382.802   <2e-16 ***
ORIGIN_SZ156      14.4620608  0.0435180   332.324   <2e-16 ***
ORIGIN_SZ172      14.9911064  0.0639192   234.532   <2e-16 ***
ORIGIN_SZ174      13.9885171  0.0470199   297.502   <2e-16 ***
ORIGIN_SZ175      16.0977944  0.0103197  1559.915   <2e-16 ***
ORIGIN_SZ176      17.2536974  0.0062502  2760.506   <2e-16 ***
ORIGIN_SZ195      13.0764242  0.0781080   167.415   <2e-16 ***
ORIGIN_SZ196      13.6161083  0.0386532   352.263   <2e-16 ***
ORIGIN_SZ197      12.5024177  0.1240488   100.786   <2e-16 ***
ORIGIN_SZ215      13.9064176  0.0864108   160.934   <2e-16 ***
ORIGIN_SZ216      17.5183224  0.0053407  3280.184   <2e-16 ***
ORIGIN_SZ217      14.8690446  0.0212652   699.221   <2e-16 ***
ORIGIN_SZ237      12.5225365  0.1666773    75.130   <2e-16 ***
ORIGIN_SZ238      13.2478875  0.0712749   185.870   <2e-16 ***
ORIGIN_SZ239      13.5878274  0.1360979    99.839   <2e-16 ***
ORIGIN_SZ257      13.5014239  0.0577665   233.724   <2e-16 ***
ORIGIN_SZ258      14.4462515  0.0285670   505.696   <2e-16 ***
ORIGIN_SZ259      14.2502358  0.0396690   359.228   <2e-16 ***
ORIGIN_SZ278      15.2806884  0.0346581   440.899   <2e-16 ***
ORIGIN_SZ279      13.2379013  0.0441496   299.842   <2e-16 ***
ORIGIN_SZ280      13.6300652  0.0479320   284.363   <2e-16 ***
ORIGIN_SZ298      10.8564638  0.3333387    32.569   <2e-16 ***
ORIGIN_SZ299      13.3702118  0.0523537   255.382   <2e-16 ***
ORIGIN_SZ300      16.5175377  0.0085691  1927.569   <2e-16 ***
ORIGIN_SZ320      13.8490521  0.0489461   282.945   <2e-16 ***
ORIGIN_SZ321      13.8203889  0.0747682   184.843   <2e-16 ***
ORIGIN_SZ322      14.7251265  0.0400786   367.406   <2e-16 ***
ORIGIN_SZ340      15.1031677  0.0271520   556.245   <2e-16 ***
ORIGIN_SZ341      13.6159022  0.0449413   302.971   <2e-16 ***
ORIGIN_SZ342      14.3919596  0.0401117   358.797   <2e-16 ***
ORIGIN_SZ363      15.1936277  0.0371860   408.585   <2e-16 ***
ORIGIN_SZ364      14.9481077  0.0265721   562.548   <2e-16 ***
ORIGIN_SZ383      14.4593095  0.0306462   471.813   <2e-16 ***
ORIGIN_SZ384      13.3345107  0.0488324   273.067   <2e-16 ***
ORIGIN_SZ385      14.6340720  0.0465590   314.313   <2e-16 ***
ORIGIN_SZ404      15.3464320  0.0290339   528.569   <2e-16 ***
ORIGIN_SZ405      14.1098214  0.0514680   274.148   <2e-16 ***
ORIGIN_SZ406      18.4385147  0.0030793  5987.858   <2e-16 ***
ORIGIN_SZ407      14.9931251  0.0390034   384.405   <2e-16 ***
ORIGIN_SZ408      15.6956577  0.0197654   794.099   <2e-16 ***
ORIGIN_SZ425      12.6447000  0.0966917   130.773   <2e-16 ***
ORIGIN_SZ426      14.4895031  0.0302644   478.763   <2e-16 ***
ORIGIN_SZ427      13.3984445  0.0423405   316.445   <2e-16 ***
ORIGIN_SZ428      13.6274613  0.0739488   184.282   <2e-16 ***
ORIGIN_SZ429      16.1986156  0.0173204   935.232   <2e-16 ***
ORIGIN_SZ446      14.1674529  0.0583537   242.786   <2e-16 ***
ORIGIN_SZ447      13.7207898  0.0402379   340.991   <2e-16 ***
ORIGIN_SZ448      14.5657430  0.0270316   538.842   <2e-16 ***
ORIGIN_SZ449      17.3659003  0.0062471  2779.836   <2e-16 ***
ORIGIN_SZ450      15.5758092  0.0174042   894.948   <2e-16 ***
ORIGIN_SZ468      14.6220318  0.0306029   477.798   <2e-16 ***
ORIGIN_SZ469      15.4969625  0.0151304  1024.228   <2e-16 ***
ORIGIN_SZ470      17.8317688  0.0046273  3853.623   <2e-16 ***
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ORIGIN_SZ1610     13.6168052  0.0599002   227.325   <2e-16 ***
ORIGIN_SZ1622     15.5684512  0.0192894   807.098   <2e-16 ***
ORIGIN_SZ1623     16.9871070  0.0040423  4202.363   <2e-16 ***
ORIGIN_SZ1624     15.8705496  0.0077461  2048.834   <2e-16 ***
ORIGIN_SZ1625     17.3243503  0.0045069  3843.988   <2e-16 ***
ORIGIN_SZ1626     17.6887405  0.0033948  5210.465   <2e-16 ***
ORIGIN_SZ1627     16.1426808  0.0068827  2345.384   <2e-16 ***
ORIGIN_SZ1628     17.8056139  0.0034587  5148.111   <2e-16 ***
ORIGIN_SZ1629     16.6865603  0.0065231  2558.060   <2e-16 ***
ORIGIN_SZ1630     16.5504558  0.0073972  2237.402   <2e-16 ***
ORIGIN_SZ1631     13.1002262  0.0534129   245.263   <2e-16 ***
ORIGIN_SZ1642     14.6718453  0.0257506   569.768   <2e-16 ***
ORIGIN_SZ1643     16.9062576  0.0043059  3926.329   <2e-16 ***
ORIGIN_SZ1644     14.8400173  0.0156911   945.757   <2e-16 ***
ORIGIN_SZ1645     16.4385774  0.0058604  2805.029   <2e-16 ***
ORIGIN_SZ1646     16.4916018  0.0084350  1955.139   <2e-16 ***
ORIGIN_SZ1647     16.5411669  0.0056062  2950.530   <2e-16 ***
ORIGIN_SZ1648     16.9237665  0.0047351  3574.116   <2e-16 ***
ORIGIN_SZ1649     17.6381005  0.0037027  4763.587   <2e-16 ***
ORIGIN_SZ1650     17.3816756  0.0050380  3450.097   <2e-16 ***
ORIGIN_SZ1664     11.9741783  0.1280491    93.512   <2e-16 ***
ORIGIN_SZ1665     17.1106006  0.0039540  4327.424   <2e-16 ***
ORIGIN_SZ1666     15.9050881  0.0067349  2361.584   <2e-16 ***
ORIGIN_SZ1667     17.3001198  0.0095058  1819.946   <2e-16 ***
ORIGIN_SZ1668     17.1391739  0.0045634  3755.789   <2e-16 ***
ORIGIN_SZ1670     17.7983089  0.0034682  5131.816   <2e-16 ***
ORIGIN_SZ1671     18.4284779  0.0049182  3746.974   <2e-16 ***
ORIGIN_SZ1672     17.5547093  0.0055162  3182.368   <2e-16 ***
ORIGIN_SZ1684     16.1794371  0.0120047  1347.755   <2e-16 ***
ORIGIN_SZ1685     16.7640475  0.0051336  3265.562   <2e-16 ***
ORIGIN_SZ1686     16.6090413  0.0051174  3245.625   <2e-16 ***
ORIGIN_SZ1687     16.4413629  0.0071544  2298.089   <2e-16 ***
ORIGIN_SZ1688     15.7900998  0.0082098  1923.320   <2e-16 ***
ORIGIN_SZ1689     15.3978751  0.0118251  1302.136   <2e-16 ***
ORIGIN_SZ1690     16.8090055  0.0063374  2652.356   <2e-16 ***
ORIGIN_SZ1691     17.3775853  0.0043159  4026.382   <2e-16 ***
ORIGIN_SZ1692     16.6775678  0.0077830  2142.818   <2e-16 ***
ORIGIN_SZ1706     16.7836218  0.0059551  2818.360   <2e-16 ***
ORIGIN_SZ1707     16.6350135  0.0049298  3374.369   <2e-16 ***
ORIGIN_SZ1708     17.2508018  0.0040396  4270.439   <2e-16 ***
ORIGIN_SZ1709     16.2724509  0.0063955  2544.340   <2e-16 ***
ORIGIN_SZ1710     17.4010425  0.0041710  4171.893   <2e-16 ***
ORIGIN_SZ1711     17.1151176  0.0048203  3550.629   <2e-16 ***
ORIGIN_SZ1712     17.7523863  0.0034462  5151.290   <2e-16 ***
ORIGIN_SZ1713     15.2754975  0.0123435  1237.534   <2e-16 ***
ORIGIN_SZ1714     17.6417565  0.0045736  3857.298   <2e-16 ***
ORIGIN_SZ1727     16.9382535  0.0050519  3352.872   <2e-16 ***
ORIGIN_SZ1728     16.8349774  0.0046638  3609.674   <2e-16 ***
ORIGIN_SZ1729     16.4379953  0.0055562  2958.474   <2e-16 ***
ORIGIN_SZ1730     14.3091948  0.0213742   669.462   <2e-16 ***
ORIGIN_SZ1731     17.0784975  0.0048745  3503.676   <2e-16 ***
ORIGIN_SZ1732     17.4923882  0.0037372  4680.599   <2e-16 ***
ORIGIN_SZ1733     17.2449149  0.0044223  3899.507   <2e-16 ***
ORIGIN_SZ1734     17.4394205  0.0042726  4081.662   <2e-16 ***
ORIGIN_SZ1735     18.3435387  0.0056002  3275.501   <2e-16 ***
ORIGIN_SZ1748     15.7374058  0.0092310  1704.848   <2e-16 ***
ORIGIN_SZ1749     16.6191733  0.0049644  3347.683   <2e-16 ***
ORIGIN_SZ1750     15.8168547  0.0071297  2218.439   <2e-16 ***
ORIGIN_SZ1751     15.4217949  0.0107640  1432.718   <2e-16 ***
ORIGIN_SZ1753     17.5468833  0.0039098  4487.880   <2e-16 ***
ORIGIN_SZ1754     17.6972890  0.0034313  5157.543   <2e-16 ***
ORIGIN_SZ1755     17.2231622  0.0043561  3953.772   <2e-16 ***
ORIGIN_SZ1756     17.1344334  0.0051967  3297.195   <2e-16 ***
ORIGIN_SZ1757     14.4418847  0.0306002   471.953   <2e-16 ***
ORIGIN_SZ1769     16.5168150  0.0057811  2857.038   <2e-16 ***
ORIGIN_SZ1770     16.7371929  0.0079885  2095.172   <2e-16 ***
ORIGIN_SZ1771     15.4061259  0.0100259  1536.631   <2e-16 ***
ORIGIN_SZ1772     14.9387889  0.0230383   648.433   <2e-16 ***
ORIGIN_SZ1774     17.2245234  0.0042829  4021.700   <2e-16 ***
ORIGIN_SZ1775     16.8435253  0.0053570  3144.204   <2e-16 ***
ORIGIN_SZ1776     17.7618706  0.0035089  5061.945   <2e-16 ***
ORIGIN_SZ1777     17.8420282  0.0039340  4535.376   <2e-16 ***
ORIGIN_SZ1778     17.0280275  0.0107762  1580.153   <2e-16 ***
ORIGIN_SZ1790     17.1652660  0.0048467  3541.618   <2e-16 ***
ORIGIN_SZ1791     16.5715530  0.0061463  2696.200   <2e-16 ***
ORIGIN_SZ1792     16.6383030  0.0064223  2590.724   <2e-16 ***
ORIGIN_SZ1793     14.9107739  0.0130230  1144.957   <2e-16 ***
ORIGIN_SZ1794     15.1350113  0.0264138   572.996   <2e-16 ***
ORIGIN_SZ1795     13.9625026  0.0319919   436.439   <2e-16 ***
ORIGIN_SZ1796     17.5465320  0.0043069  4074.010   <2e-16 ***
ORIGIN_SZ1797     17.5419624  0.0039411  4450.994   <2e-16 ***
ORIGIN_SZ1798     17.7927375  0.0039366  4519.835   <2e-16 ***
ORIGIN_SZ1799     16.5179098  0.0073993  2232.347   <2e-16 ***
ORIGIN_SZ1800     15.6593138  0.0240942   649.921   <2e-16 ***
ORIGIN_SZ1811     16.3197425  0.0070289  2321.792   <2e-16 ***
ORIGIN_SZ1812     17.3908591  0.0036798  4726.064   <2e-16 ***
ORIGIN_SZ1813     17.2743168  0.0039564  4366.153   <2e-16 ***
ORIGIN_SZ1817     17.5077879  0.0048037  3644.661   <2e-16 ***
ORIGIN_SZ1818     17.3783604  0.0043780  3969.483   <2e-16 ***
ORIGIN_SZ1819     17.6555705  0.0037319  4730.967   <2e-16 ***
ORIGIN_SZ1820     14.6650258  0.0335896   436.595   <2e-16 ***
ORIGIN_SZ1832     17.0233126  0.0051661  3295.168   <2e-16 ***
ORIGIN_SZ1833     16.4774393  0.0058075  2837.289   <2e-16 ***
ORIGIN_SZ1834     16.9567665  0.0045705  3710.035   <2e-16 ***
ORIGIN_SZ1835     16.2500715  0.0072461  2242.591   <2e-16 ***
ORIGIN_SZ1837     13.4702642  0.0605534   222.453   <2e-16 ***
ORIGIN_SZ1839     16.6922489  0.0070682  2361.591   <2e-16 ***
ORIGIN_SZ1840     18.6247050  0.0029654  6280.725   <2e-16 ***
ORIGIN_SZ1841     14.2830286  0.0350583   407.407   <2e-16 ***
ORIGIN_SZ1842     17.6316811  0.0100041  1762.451   <2e-16 ***
ORIGIN_SZ1853     16.8645635  0.0049093  3435.219   <2e-16 ***
ORIGIN_SZ1854     17.1336205  0.0046436  3689.716   <2e-16 ***
ORIGIN_SZ1855     17.4326684  0.0041866  4163.967   <2e-16 ***
ORIGIN_SZ1858     15.1207205  0.0248931   607.427   <2e-16 ***
ORIGIN_SZ1860     18.0466784  0.0080342  2246.236   <2e-16 ***
ORIGIN_SZ1861     17.6109889  0.0045070  3907.456   <2e-16 ***
ORIGIN_SZ1874     17.1205068  0.0058289  2937.193   <2e-16 ***
ORIGIN_SZ1875     15.5038243  0.0124087  1249.431   <2e-16 ***
ORIGIN_SZ1876     17.5652661  0.0096363  1822.831   <2e-16 ***
ORIGIN_SZ1877     17.0749434  0.0047545  3591.352   <2e-16 ***
ORIGIN_SZ1880     13.7901276  0.0464638   296.793   <2e-16 ***
ORIGIN_SZ1882     17.4611984  0.0048606  3592.393   <2e-16 ***
ORIGIN_SZ1883     17.5122889  0.0082642  2119.063   <2e-16 ***
ORIGIN_SZ1895     17.1175452  0.0047201  3626.505   <2e-16 ***
ORIGIN_SZ1896     15.6445239  0.0093313  1676.562   <2e-16 ***
ORIGIN_SZ1897     16.4281615  0.0084632  1941.126   <2e-16 ***
ORIGIN_SZ1898     11.9659945  0.1231079    97.199   <2e-16 ***
ORIGIN_SZ1901     13.5176969  0.0710961   190.133   <2e-16 ***
ORIGIN_SZ1903     16.8906898  0.0083617  2019.999   <2e-16 ***
ORIGIN_SZ1917     15.8732862  0.0089910  1765.470   <2e-16 ***
ORIGIN_SZ1918     17.7244187  0.0055324  3203.776   <2e-16 ***
ORIGIN_SZ1919     17.0563595  0.0050799  3357.641   <2e-16 ***
ORIGIN_SZ1922     14.8707772  0.0273452   543.817   <2e-16 ***
ORIGIN_SZ1924     16.8453980  0.0089574  1880.617   <2e-16 ***
ORIGIN_SZ1937     16.5511451  0.0066571  2486.250   <2e-16 ***
ORIGIN_SZ1938     17.3455486  0.0043828  3957.627   <2e-16 ***
ORIGIN_SZ1939     18.0078675  0.0040139  4486.418   <2e-16 ***
ORIGIN_SZ1942     14.0925286  0.0403380   349.361   <2e-16 ***
ORIGIN_SZ1959     16.2200261  0.0082820  1958.465   <2e-16 ***
ORIGIN_SZ1960     18.0723157  0.0030375  5949.683   <2e-16 ***
ORIGIN_SZ1961     16.8535335  0.0067221  2507.201   <2e-16 ***
ORIGIN_SZ1962     17.7409672  0.0043092  4116.982   <2e-16 ***
ORIGIN_SZ1964     13.2553789  0.0824997   160.672   <2e-16 ***
ORIGIN_SZ1979     16.4881568  0.0076982  2141.828   <2e-16 ***
ORIGIN_SZ1980     14.9708840  0.0136317  1098.240   <2e-16 ***
ORIGIN_SZ1981     17.4290066  0.0042820  4070.277   <2e-16 ***
ORIGIN_SZ1982     16.1847782  0.0106877  1514.338   <2e-16 ***
ORIGIN_SZ1983     17.7131611  0.0043282  4092.456   <2e-16 ***
ORIGIN_SZ1984     16.3401353  0.0074844  2183.220   <2e-16 ***
ORIGIN_SZ1985     16.9405376  0.0061212  2767.505   <2e-16 ***
ORIGIN_SZ2001     16.7667188  0.0057133  2934.668   <2e-16 ***
ORIGIN_SZ2002     17.4727864  0.0038369  4553.877   <2e-16 ***
ORIGIN_SZ2003     17.0717267  0.0049328  3460.869   <2e-16 ***
ORIGIN_SZ2004     17.8259362  0.0041105  4336.711   <2e-16 ***
ORIGIN_SZ2005     17.5889461  0.0041274  4261.480   <2e-16 ***
ORIGIN_SZ2006     16.1202663  0.0090163  1787.904   <2e-16 ***
ORIGIN_SZ2007     15.4587971  0.0158466   975.526   <2e-16 ***
ORIGIN_SZ2022     17.4262785  0.0056351  3092.458   <2e-16 ***
ORIGIN_SZ2023     17.6779653  0.0037277  4742.336   <2e-16 ***
ORIGIN_SZ2024     17.0342910  0.0047127  3614.563   <2e-16 ***
ORIGIN_SZ2025     17.2883254  0.0045580  3792.953   <2e-16 ***
ORIGIN_SZ2026     15.7631450  0.0112095  1406.230   <2e-16 ***
ORIGIN_SZ2027     17.7824851  0.0039759  4472.618   <2e-16 ***
ORIGIN_SZ2043     16.5064888  0.0074455  2216.968   <2e-16 ***
ORIGIN_SZ2044     17.4164845  0.0042103  4136.644   <2e-16 ***
ORIGIN_SZ2045     13.5484660  0.0644455   210.231   <2e-16 ***
ORIGIN_SZ2046     17.4220788  0.0039175  4447.205   <2e-16 ***
ORIGIN_SZ2047     17.3312549  0.0051236  3382.660   <2e-16 ***
ORIGIN_SZ2048     17.4222120  0.0045355  3841.271   <2e-16 ***
ORIGIN_SZ2049     16.2697515  0.0128016  1270.915   <2e-16 ***
ORIGIN_SZ2064     17.3422374  0.0047956  3616.317   <2e-16 ***
ORIGIN_SZ2065     15.9824887  0.0081983  1949.491   <2e-16 ***
ORIGIN_SZ2066     15.8922658  0.0228202   696.414   <2e-16 ***
ORIGIN_SZ2067     18.3077411  0.0028639  6392.632   <2e-16 ***
ORIGIN_SZ2068     17.5972042  0.0066806  2634.056   <2e-16 ***
ORIGIN_SZ2069     17.2219369  0.0053081  3244.459   <2e-16 ***
ORIGIN_SZ2085     16.0881664  0.0091778  1752.950   <2e-16 ***
ORIGIN_SZ2086     17.6215580  0.0039714  4437.060   <2e-16 ***
ORIGIN_SZ2087     16.3950722  0.0068826  2382.112   <2e-16 ***
ORIGIN_SZ2088     17.0908524  0.0045294  3773.308   <2e-16 ***
ORIGIN_SZ2089     16.8675590  0.0062539  2697.132   <2e-16 ***
ORIGIN_SZ2090     17.8315057  0.0034963  5100.146   <2e-16 ***
ORIGIN_SZ2091     13.6955859  0.0631468   216.885   <2e-16 ***
ORIGIN_SZ2105     14.1125812  0.0985581   143.190   <2e-16 ***
ORIGIN_SZ2106     14.6414654  0.0184780   792.373   <2e-16 ***
ORIGIN_SZ2107     15.8314290  0.0086402  1832.308   <2e-16 ***
ORIGIN_SZ2108     17.0084994  0.0053738  3165.070   <2e-16 ***
ORIGIN_SZ2109     17.2431047  0.0044610  3865.294   <2e-16 ***
ORIGIN_SZ2110     16.7916912  0.0071236  2357.179   <2e-16 ***
ORIGIN_SZ2111     14.6874255  0.0384227   382.259   <2e-16 ***
ORIGIN_SZ2128     15.6840940  0.0134195  1168.751   <2e-16 ***
ORIGIN_SZ2129     15.0891198  0.0146034  1033.258   <2e-16 ***
ORIGIN_SZ2130     17.4880134  0.0038775  4510.177   <2e-16 ***
ORIGIN_SZ2131     17.7697711  0.0043766  4060.181   <2e-16 ***
ORIGIN_SZ2132     16.9092159  0.0058602  2885.430   <2e-16 ***
ORIGIN_SZ2148     16.3905292  0.0109595  1495.556   <2e-16 ***
ORIGIN_SZ2149     15.2995992  0.0145826  1049.169   <2e-16 ***
ORIGIN_SZ2150     17.5091513  0.0044395  3943.917   <2e-16 ***
ORIGIN_SZ2151     17.7314741  0.0035096  5052.291   <2e-16 ***
ORIGIN_SZ2152     17.9129001  0.0037585  4765.994   <2e-16 ***
ORIGIN_SZ2153     17.3432375  0.0053764  3225.829   <2e-16 ***
ORIGIN_SZ2171     16.4568918  0.0068831  2390.905   <2e-16 ***
ORIGIN_SZ2172     16.6154908  0.0075332  2205.632   <2e-16 ***
ORIGIN_SZ2173     16.9974571  0.0050314  3378.252   <2e-16 ***
ORIGIN_SZ2174     17.2830673  0.0051222  3374.153   <2e-16 ***
ORIGIN_SZ2191     16.2413756  0.0097908  1658.837   <2e-16 ***
ORIGIN_SZ2192     15.8054056  0.0106206  1488.190   <2e-16 ***
ORIGIN_SZ2193     16.8469406  0.0058486  2880.520   <2e-16 ***
ORIGIN_SZ2194     17.0847071  0.0050136  3407.670   <2e-16 ***
ORIGIN_SZ2195     16.5157113  0.0199754   826.803   <2e-16 ***
ORIGIN_SZ2212     14.0634543  0.0825044   170.457   <2e-16 ***
ORIGIN_SZ2213     14.0099431  0.0361596   387.447   <2e-16 ***
ORIGIN_SZ2214     14.8668683  0.0261041   569.523   <2e-16 ***
ORIGIN_SZ2215     16.7813055  0.0064734  2592.339   <2e-16 ***
ORIGIN_SZ2216     15.8040514  0.0106287  1486.925   <2e-16 ***
ORIGIN_SZ2233     14.1499603  0.0438945   322.363   <2e-16 ***
ORIGIN_SZ2234     16.3120143  0.0191588   851.412   <2e-16 ***
ORIGIN_SZ2235     15.8017816  0.0123113  1283.516   <2e-16 ***
ORIGIN_SZ2236     15.6094875  0.0125479  1243.990   <2e-16 ***
ORIGIN_SZ2237     13.3065457  0.0559328   237.902   <2e-16 ***
ORIGIN_SZ2256     14.2009275  0.0418181   339.588   <2e-16 ***
ORIGIN_SZ2257     15.2607057  0.0188763   808.459   <2e-16 ***
ORIGIN_SZ2258     14.8781858  0.0154325   964.084   <2e-16 ***
ORIGIN_SZ2259     14.5573336  0.0330605   440.324   <2e-16 ***
ORIGIN_SZ2277     15.2450083  0.0366627   415.818   <2e-16 ***
ORIGIN_SZ2278     16.2711543  0.0113003  1439.893   <2e-16 ***
ORIGIN_SZ2279     14.6778939  0.0195616   750.342   <2e-16 ***
ORIGIN_SZ2280     12.6841698  0.1031584   122.958   <2e-16 ***
ORIGIN_SZ2297     17.2675039  0.0088944  1941.382   <2e-16 ***
ORIGIN_SZ2300     13.1130468  0.0674481   194.417   <2e-16 ***
ORIGIN_SZ2301     14.8744683  0.0182382   815.568   <2e-16 ***
ORIGIN_SZ2318     15.6275373  0.0179566   870.293   <2e-16 ***
ORIGIN_SZ2319     16.6558082  0.0099014  1682.170   <2e-16 ***
ORIGIN_SZ2322     16.1067660  0.0112754  1428.491   <2e-16 ***
ORIGIN_SZ2337     17.2852700  0.0146577  1179.264   <2e-16 ***
ORIGIN_SZ2341     16.9876357  0.0088480  1919.947   <2e-16 ***
ORIGIN_SZ2343     15.6464859  0.0135032  1158.721   <2e-16 ***
ORIGIN_SZ2361     16.6635165  0.0112716  1478.363   <2e-16 ***
ORIGIN_SZ2364     12.9334479  0.0733497   176.326   <2e-16 ***
ORIGIN_SZ2379     17.1237880  0.0162800  1051.832   <2e-16 ***
ORIGIN_SZ2384     16.2321989  0.0143144  1133.975   <2e-16 ***
ORIGIN_SZ2405     15.9593360  0.0143219  1114.335   <2e-16 ***
ORIGIN_SZ2406     14.5887345  0.0323372   451.144   <2e-16 ***
ORIGIN_SZ2426     16.3970030  0.0256370   639.583   <2e-16 ***
ORIGIN_SZ2427     16.9094915  0.0104303  1621.196   <2e-16 ***
ORIGIN_SZ2505     16.6423985  0.0384692   432.616   <2e-16 ***
log(BUSINESS_num)  0.1145699  0.0001897   604.087   <2e-16 ***
log(FB_num)        0.1319546  0.0002589   509.608   <2e-16 ***
log(schools_num)   0.1776087  0.0006586   269.689   <2e-16 ***
log(hdb_num)       0.1227611  0.0001474   832.701   <2e-16 ***
log(entertn_num)   0.2532727  0.0007717   328.190   <2e-16 ***
log(dist)         -1.4416487  0.0002609 -5526.339   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 349181331  on 64962  degrees of freedom
Residual deviance:  42866101  on 64139  degrees of freedom
AIC: 43226117

Number of Fisher Scoring iterations: 7

9.4.3 Destin Constrained Model

The code chunk used to calibrate to model is shown below is the destination:

decSIM  <- glm(formula = MORNING_PEAK ~ 
                DESTIN_SZ +
                log(BUSINESS_num) +
                log(FB_num) +
                
                log(schools_num) +
                log(hdb_num) +
                log(entertn_num)+
                log(dist)-1,
              family = poisson(link = "log"),
              data = inter_zonal_flow,
              na.action = na.exclude)

the calculated R-squared is 0.3053515, indicating that the model accounts for approximately 30.54% of the variability in the observed MORNING_PEAK.

Code
CalcRSquared(decSIM$data$MORNING_PEAK, decSIM$fitted.values)
[1] 0.3053515
Code
decSIM$coefficients[820:825]
log(BUSINESS_num)       log(FB_num)  log(schools_num)      log(hdb_num) 
               NA                NA                NA                NA 
 log(entertn_num)         log(dist) 
               NA         -1.461883 
Code
options(max.print = 10000)
summary(decSIM)

Call:
glm(formula = MORNING_PEAK ~ DESTIN_SZ + log(BUSINESS_num) + 
    log(FB_num) + log(schools_num) + log(hdb_num) + log(entertn_num) + 
    log(dist) - 1, family = poisson(link = "log"), data = inter_zonal_flow, 
    na.action = na.exclude)

Coefficients: (5 not defined because of singularities)
                    Estimate Std. Error  z value Pr(>|z|)    
DESTIN_SZ23       16.6350037  0.0504877   329.49   <2e-16 ***
DESTIN_SZ44       17.7648906  0.0226652   783.79   <2e-16 ***
DESTIN_SZ46       16.8708169  0.0191462   881.16   <2e-16 ***
DESTIN_SZ66       15.9389708  0.0461679   345.24   <2e-16 ***
DESTIN_SZ67       16.7983647  0.0137218  1224.21   <2e-16 ***
DESTIN_SZ68       16.8163886  0.0193076   870.97   <2e-16 ***
DESTIN_SZ86       16.8940255  0.0344631   490.21   <2e-16 ***
DESTIN_SZ87       13.9580875  0.0836436   166.88   <2e-16 ***
DESTIN_SZ88       16.2107274  0.0171040   947.77   <2e-16 ***
DESTIN_SZ89       16.6972232  0.0199974   834.97   <2e-16 ***
DESTIN_SZ90       12.1872374  0.7071097    17.23   <2e-16 ***
DESTIN_SZ109      15.9090781  0.0240970   660.21   <2e-16 ***
DESTIN_SZ110      13.0822409  0.1005198   130.15   <2e-16 ***
DESTIN_SZ111      16.7511015  0.0118668  1411.60   <2e-16 ***
DESTIN_SZ112      14.9617633  0.0323917   461.90   <2e-16 ***
DESTIN_SZ128      16.6176074  0.0231683   717.25   <2e-16 ***
DESTIN_SZ129      13.5696912  0.1324650   102.44   <2e-16 ***
DESTIN_SZ130      15.9680273  0.0219343   728.00   <2e-16 ***
DESTIN_SZ131      15.4024371  0.0217725   707.43   <2e-16 ***
DESTIN_SZ132      16.6387249  0.0146895  1132.69   <2e-16 ***
DESTIN_SZ133      16.4221307  0.0123079  1334.28   <2e-16 ***
DESTIN_SZ134      16.6149515  0.0115366  1440.19   <2e-16 ***
DESTIN_SZ150      16.0520048  0.0253139   634.12   <2e-16 ***
DESTIN_SZ151      15.7327201  0.0222516   707.04   <2e-16 ***
DESTIN_SZ152      17.1756622  0.0142121  1208.53   <2e-16 ***
DESTIN_SZ153      16.9664862  0.0129691  1308.23   <2e-16 ***
DESTIN_SZ154      15.1305365  0.0271986   556.30   <2e-16 ***
DESTIN_SZ155      15.8372324  0.0184088   860.31   <2e-16 ***
DESTIN_SZ156      16.3358762  0.0196674   830.61   <2e-16 ***
DESTIN_SZ172      14.5845392  0.0430694   338.63   <2e-16 ***
DESTIN_SZ174      16.4021989  0.0143354  1144.18   <2e-16 ***
DESTIN_SZ175      16.5471268  0.0104815  1578.70   <2e-16 ***
DESTIN_SZ176      14.9585939  0.0264885   564.72   <2e-16 ***
DESTIN_SZ195      15.7298146  0.0295692   531.97   <2e-16 ***
DESTIN_SZ196      16.6844551  0.0100408  1661.67   <2e-16 ***
DESTIN_SZ197      13.4886576  0.0751863   179.40   <2e-16 ***
DESTIN_SZ215      16.5514188  0.0290274   570.20   <2e-16 ***
DESTIN_SZ216      15.8626889  0.0152046  1043.28   <2e-16 ***
DESTIN_SZ217      17.2783169  0.0091383  1890.76   <2e-16 ***
DESTIN_SZ237      16.1171510  0.0305242   528.01   <2e-16 ***
DESTIN_SZ238      16.6697988  0.0147529  1129.93   <2e-16 ***
DESTIN_SZ239      15.6539571  0.0301673   518.90   <2e-16 ***
DESTIN_SZ257      16.4832448  0.0154009  1070.28   <2e-16 ***
DESTIN_SZ258      15.9682169  0.0190669   837.48   <2e-16 ***
DESTIN_SZ259      17.1908916  0.0118314  1452.99   <2e-16 ***
DESTIN_SZ278      16.7794448  0.0137832  1217.38   <2e-16 ***
DESTIN_SZ279      16.4807373  0.0130646  1261.48   <2e-16 ***
DESTIN_SZ280      16.8293849  0.0122569  1373.06   <2e-16 ***
DESTIN_SZ298      13.6812668  0.1291111   105.97   <2e-16 ***
DESTIN_SZ299      16.6761994  0.0157800  1056.80   <2e-16 ***
DESTIN_SZ300      15.8888699  0.0167336   949.52   <2e-16 ***
DESTIN_SZ320      17.1120970  0.0137400  1245.42   <2e-16 ***
DESTIN_SZ321      15.8931575  0.0233389   680.97   <2e-16 ***
DESTIN_SZ322      17.9990148  0.0114968  1565.56   <2e-16 ***
DESTIN_SZ340      18.0084556  0.0105046  1714.34   <2e-16 ***
DESTIN_SZ341      16.8407452  0.0119767  1406.13   <2e-16 ***
DESTIN_SZ342      17.8472835  0.0102467  1741.75   <2e-16 ***
DESTIN_SZ363      17.0920934  0.0130186  1312.90   <2e-16 ***
DESTIN_SZ364      18.1472210  0.0076257  2379.75   <2e-16 ***
DESTIN_SZ383      17.1539570  0.0087594  1958.34   <2e-16 ***
DESTIN_SZ384      18.2281528  0.0082303  2214.77   <2e-16 ***
DESTIN_SZ385      17.2050842  0.0145717  1180.72   <2e-16 ***
DESTIN_SZ404      17.8057962  0.0123818  1438.07   <2e-16 ***
DESTIN_SZ405      17.2056191  0.0102264  1682.48   <2e-16 ***
DESTIN_SZ406      17.5162432  0.0051447  3404.71   <2e-16 ***
DESTIN_SZ407      18.8769281  0.0076649  2462.76   <2e-16 ***
DESTIN_SZ408      19.3525187  0.0043373  4461.89   <2e-16 ***
DESTIN_SZ425      15.7913467  0.0243422   648.72   <2e-16 ***
DESTIN_SZ426      17.0046451  0.0100827  1686.52   <2e-16 ***
DESTIN_SZ427      15.0595385  0.0227629   661.58   <2e-16 ***
DESTIN_SZ428      17.4326472  0.0096489  1806.69   <2e-16 ***
DESTIN_SZ429      16.7217277  0.0212361   787.42   <2e-16 ***
DESTIN_SZ446      17.1256672  0.0223256   767.09   <2e-16 ***
DESTIN_SZ447      16.0327727  0.0168113   953.69   <2e-16 ***
DESTIN_SZ448      17.0192011  0.0114702  1483.78   <2e-16 ***
DESTIN_SZ449      17.7883748  0.0069594  2556.02   <2e-16 ***
DESTIN_SZ450      16.9471362  0.0110126  1538.89   <2e-16 ***
DESTIN_SZ468      17.2399857  0.0105525  1633.73   <2e-16 ***
DESTIN_SZ469      17.1924367  0.0083100  2068.88   <2e-16 ***
DESTIN_SZ470      17.5791594  0.0065198  2696.29   <2e-16 ***
DESTIN_SZ471      16.5772156  0.0152953  1083.81   <2e-16 ***
DESTIN_SZ488      16.1865316  0.0217661   743.66   <2e-16 ***
DESTIN_SZ489      14.7212952  0.0457261   321.94   <2e-16 ***
DESTIN_SZ490      17.3463466  0.0084917  2042.74   <2e-16 ***
DESTIN_SZ491      15.9770695  0.0148026  1079.34   <2e-16 ***
DESTIN_SZ493      14.3167122  0.0477080   300.09   <2e-16 ***
DESTIN_SZ494      16.2070130  0.0216343   749.14   <2e-16 ***
DESTIN_SZ509      15.9972914  0.0147449  1084.94   <2e-16 ***
DESTIN_SZ510      17.3356293  0.0096472  1796.97   <2e-16 ***
DESTIN_SZ511      17.2671211  0.0066432  2599.22   <2e-16 ***
DESTIN_SZ512      17.3427742  0.0063618  2726.09   <2e-16 ***
DESTIN_SZ513      17.7464204  0.0072148  2459.71   <2e-16 ***
DESTIN_SZ514      16.9224595  0.0116174  1456.64   <2e-16 ***
DESTIN_SZ515      15.6696805  0.0296989   527.62   <2e-16 ***
DESTIN_SZ530      15.3908524  0.0283953   542.02   <2e-16 ***
DESTIN_SZ531      17.9922198  0.0058826  3058.57   <2e-16 ***
DESTIN_SZ532      16.9617561  0.0087074  1947.97   <2e-16 ***
DESTIN_SZ533      18.9405165  0.0031746  5966.21   <2e-16 ***
DESTIN_SZ534      17.6841769  0.0063900  2767.49   <2e-16 ***
DESTIN_SZ536      15.8542723  0.0175085   905.52   <2e-16 ***
DESTIN_SZ537      15.3354540  0.0333316   460.09   <2e-16 ***
DESTIN_SZ538      15.0159050  0.0426815   351.81   <2e-16 ***
DESTIN_SZ539      14.0001197  0.2500093    56.00   <2e-16 ***
DESTIN_SZ551      16.3176984  0.0173472   940.65   <2e-16 ***
DESTIN_SZ552      18.6362388  0.0103292  1804.23   <2e-16 ***
DESTIN_SZ553      16.9884919  0.0084946  1999.93   <2e-16 ***
DESTIN_SZ554      17.3254481  0.0066826  2592.61   <2e-16 ***
DESTIN_SZ555      17.6793305  0.0073795  2395.72   <2e-16 ***
DESTIN_SZ559      15.6614521  0.0529668   295.68   <2e-16 ***
DESTIN_SZ560      16.9584128  0.0161495  1050.09   <2e-16 ***
DESTIN_SZ561      15.6756427  0.0641880   244.21   <2e-16 ***
DESTIN_SZ572      16.5303733  0.0342533   482.59   <2e-16 ***
DESTIN_SZ573      17.2474249  0.0099580  1732.02   <2e-16 ***
DESTIN_SZ574      17.1383588  0.0154267  1110.95   <2e-16 ***
DESTIN_SZ575      19.4038343  0.0025184  7704.92   <2e-16 ***
DESTIN_SZ576      17.4550171  0.0062076  2811.90   <2e-16 ***
DESTIN_SZ578      13.2867423  0.0976082   136.12   <2e-16 ***
DESTIN_SZ582      14.8272003  0.0612279   242.16   <2e-16 ***
DESTIN_SZ583      13.6634209  0.1825843    74.83   <2e-16 ***
DESTIN_SZ584      16.6164471  0.0343869   483.22   <2e-16 ***
DESTIN_SZ593      15.7980949  0.0249032   634.38   <2e-16 ***
DESTIN_SZ594      16.6145107  0.0133112  1248.16   <2e-16 ***
DESTIN_SZ595      17.6899936  0.0058536  3022.09   <2e-16 ***
DESTIN_SZ596      17.6715881  0.0050997  3465.24   <2e-16 ***
DESTIN_SZ597      17.3190976  0.0149428  1159.03   <2e-16 ***
DESTIN_SZ603      14.7314038  0.0511291   288.12   <2e-16 ***
DESTIN_SZ604      16.9938545  0.0492599   344.98   <2e-16 ***
DESTIN_SZ615      15.9197596  0.0144271  1103.46   <2e-16 ***
DESTIN_SZ616      17.4648127  0.0086819  2011.64   <2e-16 ***
DESTIN_SZ617      16.5733942  0.0090141  1838.61   <2e-16 ***
DESTIN_SZ618      17.4535085  0.0062524  2791.51   <2e-16 ***
DESTIN_SZ620      17.1285449  0.0106657  1605.95   <2e-16 ***
DESTIN_SZ637      16.9761427  0.0107006  1586.47   <2e-16 ***
DESTIN_SZ638      17.4409559  0.0057167  3050.87   <2e-16 ***
DESTIN_SZ657      17.5626225  0.0070019  2508.27   <2e-16 ***
DESTIN_SZ658      17.3208974  0.0060701  2853.49   <2e-16 ***
DESTIN_SZ659      17.6384628  0.0057368  3074.62   <2e-16 ***
DESTIN_SZ660      18.2538855  0.0040861  4467.32   <2e-16 ***
DESTIN_SZ662      16.9808432  0.0115807  1466.31   <2e-16 ***
DESTIN_SZ677      17.7386129  0.0092539  1916.89   <2e-16 ***
DESTIN_SZ678      16.1434356  0.0137714  1172.24   <2e-16 ***
DESTIN_SZ679      17.6987142  0.0049770  3556.13   <2e-16 ***
DESTIN_SZ680      19.5108225  0.0026778  7286.16   <2e-16 ***
DESTIN_SZ681      16.2941696  0.0175510   928.39   <2e-16 ***
DESTIN_SZ699      17.0802411  0.0089024  1918.61   <2e-16 ***
DESTIN_SZ700      17.3754689  0.0068593  2533.14   <2e-16 ***
DESTIN_SZ701      15.3904622  0.0191672   802.96   <2e-16 ***
DESTIN_SZ702      17.8741006  0.0046971  3805.34   <2e-16 ***
DESTIN_SZ704      17.3239918  0.0106139  1632.19   <2e-16 ***
DESTIN_SZ722      17.1982826  0.0087045  1975.80   <2e-16 ***
DESTIN_SZ725      15.6113259  0.0259626   601.30   <2e-16 ***
DESTIN_SZ741      17.6248992  0.0071347  2470.30   <2e-16 ***
DESTIN_SZ743      15.0699990  0.0203173   741.73   <2e-16 ***
DESTIN_SZ744      17.9352949  0.0048603  3690.20   <2e-16 ***
DESTIN_SZ761      17.7189488  0.0082410  2150.09   <2e-16 ***
DESTIN_SZ762      16.0792596  0.0131711  1220.80   <2e-16 ***
DESTIN_SZ763      15.1629935  0.0196043   773.45   <2e-16 ***
DESTIN_SZ764      18.2237883  0.0037249  4892.36   <2e-16 ***
DESTIN_SZ765      17.3932101  0.0095007  1830.74   <2e-16 ***
DESTIN_SZ767      17.6473791  0.0066086  2670.35   <2e-16 ***
DESTIN_SZ772      15.2976079  0.0378704   403.95   <2e-16 ***
DESTIN_SZ784      15.3332579  0.0209105   733.28   <2e-16 ***
DESTIN_SZ785      17.1377007  0.0055825  3069.88   <2e-16 ***
DESTIN_SZ786      17.6152552  0.0050562  3483.89   <2e-16 ***
DESTIN_SZ787      16.9017027  0.0106515  1586.79   <2e-16 ***
DESTIN_SZ788      16.3233043  0.0141549  1153.19   <2e-16 ***
DESTIN_SZ789      16.3678937  0.0168129   973.53   <2e-16 ***
DESTIN_SZ803      16.7814124  0.0138660  1210.26   <2e-16 ***
DESTIN_SZ804      17.3973577  0.0062360  2789.81   <2e-16 ***
DESTIN_SZ805      18.5906637  0.0031242  5950.59   <2e-16 ***
DESTIN_SZ806      17.3213007  0.0057078  3034.65   <2e-16 ***
DESTIN_SZ807      17.6772834  0.0067598  2615.05   <2e-16 ***
DESTIN_SZ808      15.9582828  0.0174760   913.15   <2e-16 ***
DESTIN_SZ809      17.6549220  0.0051956  3398.05   <2e-16 ***
DESTIN_SZ810      17.5998611  0.0079285  2219.82   <2e-16 ***
DESTIN_SZ814      16.0785490  0.0193294   831.82   <2e-16 ***
DESTIN_SZ819      21.0709878  0.0134508  1566.52   <2e-16 ***
DESTIN_SZ824      16.7187278  0.0133783  1249.69   <2e-16 ***
DESTIN_SZ826      17.1553849  0.0060271  2846.39   <2e-16 ***
DESTIN_SZ827      17.1267880  0.0063073  2715.39   <2e-16 ***
DESTIN_SZ828      17.6009511  0.0048910  3598.63   <2e-16 ***
DESTIN_SZ829      17.1555531  0.0076612  2239.27   <2e-16 ***
DESTIN_SZ830      16.3472557  0.0125195  1305.75   <2e-16 ***
DESTIN_SZ831      19.2155043  0.0027893  6889.01   <2e-16 ***
DESTIN_SZ832      19.4880316  0.0034352  5673.04   <2e-16 ***
DESTIN_SZ835      15.6053946  0.0200445   778.54   <2e-16 ***
DESTIN_SZ844      16.5887386  0.0178695   928.33   <2e-16 ***
DESTIN_SZ846      17.0205239  0.0068926  2469.38   <2e-16 ***
DESTIN_SZ847      18.0457080  0.0046336  3894.57   <2e-16 ***
DESTIN_SZ848      17.1457727  0.0063440  2702.66   <2e-16 ***
DESTIN_SZ849      16.8029926  0.0080767  2080.44   <2e-16 ***
DESTIN_SZ850      17.6241679  0.0053437  3298.12   <2e-16 ***
DESTIN_SZ851      17.2416374  0.0061193  2817.60   <2e-16 ***
DESTIN_SZ852      17.2686220  0.0079876  2161.93   <2e-16 ***
DESTIN_SZ853      18.0919533  0.0078972  2290.94   <2e-16 ***
DESTIN_SZ854      16.5200790  0.0137976  1197.32   <2e-16 ***
DESTIN_SZ855      15.4557657  0.0194006   796.66   <2e-16 ***
DESTIN_SZ856      14.8399121  0.0305945   485.05   <2e-16 ***
DESTIN_SZ866      17.1783903  0.0097036  1770.32   <2e-16 ***
DESTIN_SZ867      18.4421175  0.0046564  3960.55   <2e-16 ***
DESTIN_SZ868      18.0440450  0.0053731  3358.20   <2e-16 ***
DESTIN_SZ869      18.6166164  0.0049498  3761.08   <2e-16 ***
DESTIN_SZ870      18.7029063  0.0030627  6106.76   <2e-16 ***
DESTIN_SZ871      19.1174343  0.0038445  4972.67   <2e-16 ***
DESTIN_SZ872      17.7233331  0.0054934  3226.32   <2e-16 ***
DESTIN_SZ873      17.4924371  0.0077039  2270.59   <2e-16 ***
DESTIN_SZ874      15.8842354  0.0130317  1218.90   <2e-16 ***
DESTIN_SZ875      15.3103070  0.0265551   576.55   <2e-16 ***
DESTIN_SZ876      15.8193529  0.0180447   876.68   <2e-16 ***
DESTIN_SZ877      16.4270996  0.0137100  1198.18   <2e-16 ***
DESTIN_SZ887      17.0729027  0.0086210  1980.38   <2e-16 ***
DESTIN_SZ888      15.9761569  0.0133600  1195.82   <2e-16 ***
DESTIN_SZ889      16.4666257  0.0125381  1313.33   <2e-16 ***
DESTIN_SZ890      17.6386824  0.0049282  3579.15   <2e-16 ***
DESTIN_SZ891      16.8883777  0.0140295  1203.77   <2e-16 ***
DESTIN_SZ893      17.9271272  0.0044286  4048.08   <2e-16 ***
DESTIN_SZ894      16.2755980  0.0110545  1472.31   <2e-16 ***
DESTIN_SZ895      16.8428082  0.0089572  1880.37   <2e-16 ***
DESTIN_SZ896      14.9497968  0.0317433   470.96   <2e-16 ***
DESTIN_SZ897      14.4624238  0.0281417   513.91   <2e-16 ***
DESTIN_SZ898      16.2524384  0.0159912  1016.34   <2e-16 ***
DESTIN_SZ908      17.2608011  0.0103537  1667.12   <2e-16 ***
DESTIN_SZ909      17.0446357  0.0061166  2786.63   <2e-16 ***
DESTIN_SZ910      16.0868090  0.0114293  1407.51   <2e-16 ***
DESTIN_SZ911      18.3128498  0.0040271  4547.36   <2e-16 ***
DESTIN_SZ912      16.7870131  0.0071293  2354.66   <2e-16 ***
DESTIN_SZ915      17.5043509  0.0057772  3029.89   <2e-16 ***
DESTIN_SZ917      17.9323462  0.0055276  3244.13   <2e-16 ***
DESTIN_SZ918      14.5122858  0.0383938   377.99   <2e-16 ***
DESTIN_SZ919      15.9198694  0.0146668  1085.44   <2e-16 ***
DESTIN_SZ928      15.8618003  0.0172313   920.52   <2e-16 ***
DESTIN_SZ929      16.8507709  0.0070258  2398.41   <2e-16 ***
DESTIN_SZ930      17.9928981  0.0038114  4720.84   <2e-16 ***
DESTIN_SZ931      16.5183596  0.0101989  1619.61   <2e-16 ***
DESTIN_SZ932      15.6268439  0.0238812   654.36   <2e-16 ***
DESTIN_SZ933      17.0836573  0.0075258  2270.02   <2e-16 ***
DESTIN_SZ934      17.0591213  0.0077684  2195.95   <2e-16 ***
DESTIN_SZ935      19.1681958  0.0027704  6918.96   <2e-16 ***
DESTIN_SZ938      13.0246629  0.1360947    95.70   <2e-16 ***
DESTIN_SZ939      18.7758507  0.0036062  5206.61   <2e-16 ***
DESTIN_SZ940      16.0253708  0.0189819   844.24   <2e-16 ***
DESTIN_SZ949      15.8607497  0.0142937  1109.64   <2e-16 ***
DESTIN_SZ950      18.1245019  0.0048519  3735.54   <2e-16 ***
DESTIN_SZ951      18.7411179  0.0029341  6387.27   <2e-16 ***
DESTIN_SZ952      18.1555445  0.0075945  2390.61   <2e-16 ***
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DESTIN_SZ2047     16.5589614  0.0090044  1838.98   <2e-16 ***
DESTIN_SZ2048     16.6440793  0.0084165  1977.57   <2e-16 ***
DESTIN_SZ2049     14.9903280  0.0276919   541.33   <2e-16 ***
DESTIN_SZ2064     17.0500355  0.0069422  2455.99   <2e-16 ***
DESTIN_SZ2065     16.9370135  0.0070668  2396.70   <2e-16 ***
DESTIN_SZ2066     15.1753466  0.0324733   467.32   <2e-16 ***
DESTIN_SZ2067     19.5021151  0.0024554  7942.46   <2e-16 ***
DESTIN_SZ2068     16.6213149  0.0130997  1268.83   <2e-16 ***
DESTIN_SZ2069     16.8594945  0.0082006  2055.89   <2e-16 ***
DESTIN_SZ2085     17.0904490  0.0073605  2321.90   <2e-16 ***
DESTIN_SZ2086     18.2772357  0.0037498  4874.16   <2e-16 ***
DESTIN_SZ2087     18.0889350  0.0040279  4490.89   <2e-16 ***
DESTIN_SZ2088     17.3029530  0.0056087  3085.01   <2e-16 ***
DESTIN_SZ2089     17.0015397  0.0088051  1930.86   <2e-16 ***
DESTIN_SZ2090     19.1760482  0.0027444  6987.30   <2e-16 ***
DESTIN_SZ2091     14.0647667  0.0665418   211.37   <2e-16 ***
DESTIN_SZ2105     15.9401982  0.0636648   250.38   <2e-16 ***
DESTIN_SZ2106     15.5885926  0.0148891  1046.98   <2e-16 ***
DESTIN_SZ2107     18.2606668  0.0037141  4916.53   <2e-16 ***
DESTIN_SZ2108     18.0256039  0.0045837  3932.55   <2e-16 ***
DESTIN_SZ2109     17.0341340  0.0064513  2640.41   <2e-16 ***
DESTIN_SZ2110     16.4783325  0.0115616  1425.26   <2e-16 ***
DESTIN_SZ2111     15.1465446  0.0329430   459.78   <2e-16 ***
DESTIN_SZ2128     17.2202324  0.0082295  2092.51   <2e-16 ***
DESTIN_SZ2129     16.5528201  0.0098828  1674.91   <2e-16 ***
DESTIN_SZ2130     17.3476185  0.0053464  3244.73   <2e-16 ***
DESTIN_SZ2131     17.4105124  0.0069407  2508.48   <2e-16 ***
DESTIN_SZ2132     17.0130022  0.0074096  2296.07   <2e-16 ***
DESTIN_SZ2148     16.3085207  0.0146352  1114.34   <2e-16 ***
DESTIN_SZ2149     16.3982869  0.0108065  1517.45   <2e-16 ***
DESTIN_SZ2150     16.7223793  0.0081235  2058.51   <2e-16 ***
DESTIN_SZ2151     18.6217838  0.0031665  5880.87   <2e-16 ***
DESTIN_SZ2152     17.7368329  0.0050850  3488.08   <2e-16 ***
DESTIN_SZ2153     16.4151939  0.0111532  1471.79   <2e-16 ***
DESTIN_SZ2171     17.5000225  0.0054451  3213.91   <2e-16 ***
DESTIN_SZ2172     16.3172933  0.0109332  1492.45   <2e-16 ***
DESTIN_SZ2173     16.7742789  0.0072780  2304.78   <2e-16 ***
DESTIN_SZ2174     17.3302208  0.0065154  2659.90   <2e-16 ***
DESTIN_SZ2191     17.5527703  0.0073207  2397.67   <2e-16 ***
DESTIN_SZ2192     16.3931709  0.0107252  1528.47   <2e-16 ***
DESTIN_SZ2193     17.2515636  0.0063032  2736.94   <2e-16 ***
DESTIN_SZ2194     17.3483342  0.0056916  3048.04   <2e-16 ***
DESTIN_SZ2195     15.8344157  0.0354660   446.47   <2e-16 ***
DESTIN_SZ2212     17.5803345  0.0127446  1379.43   <2e-16 ***
DESTIN_SZ2213     17.3429779  0.0080955  2142.29   <2e-16 ***
DESTIN_SZ2214     18.0067877  0.0069758  2581.32   <2e-16 ***
DESTIN_SZ2215     16.4921760  0.0094232  1750.17   <2e-16 ***
DESTIN_SZ2216     17.5061556  0.0060488  2894.17   <2e-16 ***
DESTIN_SZ2233     18.1631439  0.0096552  1881.17   <2e-16 ***
DESTIN_SZ2234     18.2930066  0.0081262  2251.11   <2e-16 ***
DESTIN_SZ2235     17.2635477  0.0087521  1972.51   <2e-16 ***
DESTIN_SZ2236     16.0178504  0.0136219  1175.89   <2e-16 ***
DESTIN_SZ2237     16.8737028  0.0112997  1493.29   <2e-16 ***
DESTIN_SZ2256     17.3481627  0.0119974  1446.00   <2e-16 ***
DESTIN_SZ2257     14.8625823  0.0261906   567.48   <2e-16 ***
DESTIN_SZ2258     16.8838961  0.0073537  2295.97   <2e-16 ***
DESTIN_SZ2259     17.7783318  0.0102016  1742.70   <2e-16 ***
DESTIN_SZ2277     16.1709752  0.0255881   631.97   <2e-16 ***
DESTIN_SZ2278     16.2155661  0.0157684  1028.36   <2e-16 ***
DESTIN_SZ2279     16.9167526  0.0075457  2241.92   <2e-16 ***
DESTIN_SZ2280     15.3121609  0.0311673   491.29   <2e-16 ***
DESTIN_SZ2297     16.1713813  0.0185179   873.28   <2e-16 ***
DESTIN_SZ2300     15.8616765  0.0205042   773.58   <2e-16 ***
DESTIN_SZ2301     16.6383378  0.0098711  1685.55   <2e-16 ***
DESTIN_SZ2318     18.6841212  0.0057960  3223.60   <2e-16 ***
DESTIN_SZ2319     18.5967829  0.0052309  3555.19   <2e-16 ***
DESTIN_SZ2322     17.1295002  0.0091323  1875.71   <2e-16 ***
DESTIN_SZ2337     17.5512708  0.0152767  1148.89   <2e-16 ***
DESTIN_SZ2341     18.9567095  0.0051747  3663.34   <2e-16 ***
DESTIN_SZ2343     16.1664742  0.0128582  1257.29   <2e-16 ***
DESTIN_SZ2361     17.4875772  0.0092317  1894.30   <2e-16 ***
DESTIN_SZ2364     14.9083138  0.0295342   504.78   <2e-16 ***
DESTIN_SZ2379     16.6530440  0.0241213   690.39   <2e-16 ***
DESTIN_SZ2384     18.1613369  0.0066839  2717.16   <2e-16 ***
DESTIN_SZ2405     18.5765284  0.0048050  3866.10   <2e-16 ***
DESTIN_SZ2406     15.5243322  0.0235366   659.58   <2e-16 ***
DESTIN_SZ2426     17.9192434  0.0109187  1641.14   <2e-16 ***
DESTIN_SZ2427     18.1441130  0.0066972  2709.19   <2e-16 ***
DESTIN_SZ2505     18.1648361  0.0181967   998.25   <2e-16 ***
log(BUSINESS_num)         NA         NA       NA       NA    
log(FB_num)               NA         NA       NA       NA    
log(schools_num)          NA         NA       NA       NA    
log(hdb_num)              NA         NA       NA       NA    
log(entertn_num)          NA         NA       NA       NA    
log(dist)         -1.4618825  0.0002562 -5705.39   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 349181331  on 64962  degrees of freedom
Residual deviance:  45737568  on 64142  degrees of freedom
AIC: 46097578

Number of Fisher Scoring iterations: 8

9.4.4 Doubly constrained

dbcSIM <- glm(formula = MORNING_PEAK ~ ORIGIN_SZ + DESTIN_SZ + log(dist), family = poisson(link = "log"), data = inter_zonal_flow, na.action = na.exclude)

the calculated R-squared is 0.5871965, indicating that the model accounts for approximately 58.72% of the variability in the observed MORNING_PEAK.

Code
CalcRSquared(dbcSIM$data$MORNING_PEAK, dbcSIM$fitted.values)
[1] 0.5871965
Code
dbcSIM$coefficients[1]
(Intercept) 
   14.78251 
Code
dbcSIM$coefficients[1635]
DESTIN_SZ2505 
    0.7094831 
Code
options(max.print = 10000)
summary(dbcSIM)

Call:
glm(formula = MORNING_PEAK ~ ORIGIN_SZ + DESTIN_SZ + log(dist), 
    family = poisson(link = "log"), data = inter_zonal_flow, 
    na.action = na.exclude)

Coefficients:
                Estimate Std. Error   z value Pr(>|z|)    
(Intercept)   14.7825131  0.1376669   107.379  < 2e-16 ***
ORIGIN_SZ44    1.8651637  0.1913130     9.749  < 2e-16 ***
ORIGIN_SZ46   -0.4638760  0.1580502    -2.935 0.003336 ** 
ORIGIN_SZ66    0.8207510  0.1832131     4.480 7.47e-06 ***
ORIGIN_SZ67    1.7280743  0.1304829    13.244  < 2e-16 ***
ORIGIN_SZ68   -0.5924438  0.1696234    -3.493 0.000478 ***
ORIGIN_SZ86   -0.4088340  0.1503510    -2.719 0.006544 ** 
ORIGIN_SZ87    3.5035738  0.1480694    23.662  < 2e-16 ***
ORIGIN_SZ88    2.5215503  0.1292031    19.516  < 2e-16 ***
ORIGIN_SZ89    0.2300891  0.1375846     1.672 0.094456 .  
ORIGIN_SZ90    1.0781361  0.1779172     6.060 1.36e-09 ***
ORIGIN_SZ109   0.6458122  0.1567823     4.119 3.80e-05 ***
ORIGIN_SZ110   0.6774575  0.1494445     4.533 5.81e-06 ***
ORIGIN_SZ111   4.0448044  0.1288315    31.396  < 2e-16 ***
ORIGIN_SZ112   0.1521353  0.1453988     1.046 0.295408    
ORIGIN_SZ128   3.3585869  0.1312681    25.586  < 2e-16 ***
ORIGIN_SZ129   0.4780069  0.2187567     2.185 0.028881 *  
ORIGIN_SZ130   1.4046878  0.1390554    10.102  < 2e-16 ***
ORIGIN_SZ131   1.1414313  0.1322103     8.633  < 2e-16 ***
ORIGIN_SZ132   0.2245551  0.1388237     1.618 0.105758    
ORIGIN_SZ133   1.1920048  0.1344194     8.868  < 2e-16 ***
ORIGIN_SZ134   4.3098259  0.1312397    32.839  < 2e-16 ***
ORIGIN_SZ150   4.3536568  0.1297949    33.543  < 2e-16 ***
ORIGIN_SZ151   0.7972520  0.1445269     5.516 3.46e-08 ***
ORIGIN_SZ152   0.2520616  0.1527246     1.650 0.098855 .  
ORIGIN_SZ153   1.6168138  0.1350720    11.970  < 2e-16 ***
ORIGIN_SZ154   4.2669539  0.1283580    33.243  < 2e-16 ***
ORIGIN_SZ155   1.2036839  0.1329298     9.055  < 2e-16 ***
ORIGIN_SZ156   2.1952570  0.1355002    16.201  < 2e-16 ***
ORIGIN_SZ172   2.4447822  0.1435353    17.033  < 2e-16 ***
ORIGIN_SZ174   1.1354702  0.1367109     8.306  < 2e-16 ***
ORIGIN_SZ175   3.8895209  0.1286409    30.235  < 2e-16 ***
ORIGIN_SZ176   4.9927160  0.1282428    38.932  < 2e-16 ***
ORIGIN_SZ195   0.1461498  0.1503134     0.972 0.330901    
ORIGIN_SZ196   1.1205023  0.1339388     8.366  < 2e-16 ***
ORIGIN_SZ197   0.2039552  0.1785044     1.143 0.253214    
ORIGIN_SZ215   1.7228934  0.1548747    11.124  < 2e-16 ***
ORIGIN_SZ216   5.3512021  0.1283651    41.687  < 2e-16 ***
ORIGIN_SZ217   2.4696949  0.1299800    19.001  < 2e-16 ***
ORIGIN_SZ237  -0.1148070  0.2104278    -0.546 0.585349    
ORIGIN_SZ238   0.9871565  0.1467267     6.728 1.72e-11 ***
ORIGIN_SZ239   1.2506722  0.1870828     6.685 2.31e-11 ***
ORIGIN_SZ257   0.2715459  0.1408892     1.927 0.053933 .  
ORIGIN_SZ258   2.2349763  0.1314398    17.004  < 2e-16 ***
ORIGIN_SZ259   1.6038725  0.1342585    11.946  < 2e-16 ***
ORIGIN_SZ278   1.8380732  0.1332248    13.797  < 2e-16 ***
ORIGIN_SZ279   0.6102791  0.1357349     4.496 6.92e-06 ***
ORIGIN_SZ280   1.0967209  0.1369188     8.010 1.15e-15 ***
ORIGIN_SZ298  -2.9525019  0.3573065    -8.263  < 2e-16 ***
ORIGIN_SZ299   0.4052485  0.1386668     2.922 0.003473 ** 
ORIGIN_SZ300   4.2242256  0.1285551    32.859  < 2e-16 ***
ORIGIN_SZ320  -0.1304437  0.1374289    -0.949 0.342533    
ORIGIN_SZ321   1.0223496  0.1486339     6.878 6.06e-12 ***
ORIGIN_SZ322   2.1303020  0.1344205    15.848  < 2e-16 ***
ORIGIN_SZ340   1.7776512  0.1312870    13.540  < 2e-16 ***
ORIGIN_SZ341   0.1698864  0.1359914     1.249 0.211576    
ORIGIN_SZ342   2.0446107  0.1344315    15.209  < 2e-16 ***
ORIGIN_SZ363   1.7399036  0.1336962    13.014  < 2e-16 ***
ORIGIN_SZ364   2.2324348  0.1310346    17.037  < 2e-16 ***
ORIGIN_SZ383   1.2662322  0.1319317     9.598  < 2e-16 ***
ORIGIN_SZ384   0.4729125  0.1372673     3.445 0.000571 ***
ORIGIN_SZ385   1.6732365  0.1364878    12.259  < 2e-16 ***
ORIGIN_SZ404   2.2671817  0.1316880    17.216  < 2e-16 ***
ORIGIN_SZ405   0.6452881  0.1382800     4.667 3.06e-06 ***
ORIGIN_SZ406   6.0984179  0.1282803    47.540  < 2e-16 ***
ORIGIN_SZ407   2.3074859  0.1341287    17.204  < 2e-16 ***
ORIGIN_SZ408   3.0326485  0.1298347    23.358  < 2e-16 ***
ORIGIN_SZ425  -0.4946901  0.1608092    -3.076 0.002096 ** 
ORIGIN_SZ426   1.5311039  0.1318184    11.615  < 2e-16 ***
ORIGIN_SZ427   0.9190448  0.1350628     6.805 1.01e-11 ***
ORIGIN_SZ428   1.0913201  0.1480848     7.370 1.71e-13 ***
ORIGIN_SZ429   2.5227525  0.1294959    19.481  < 2e-16 ***
ORIGIN_SZ446   1.4303619  0.1412628    10.126  < 2e-16 ***
ORIGIN_SZ447   0.7386870  0.1344585     5.494 3.93e-08 ***
ORIGIN_SZ448   1.9352844  0.1310960    14.762  < 2e-16 ***
ORIGIN_SZ449   5.2559562  0.1284385    40.922  < 2e-16 ***
ORIGIN_SZ450   2.2474602  0.1294904    17.356  < 2e-16 ***
ORIGIN_SZ468   1.9333540  0.1318865    14.659  < 2e-16 ***
ORIGIN_SZ469   3.1703947  0.1291647    24.545  < 2e-16 ***
ORIGIN_SZ470   5.2270506  0.1283788    40.716  < 2e-16 ***
ORIGIN_SZ471   3.2081727  0.1291587    24.839  < 2e-16 ***
ORIGIN_SZ488   0.8657735  0.1398141     6.192 5.93e-10 ***
ORIGIN_SZ489  -0.8857821  0.2053213    -4.314 1.60e-05 ***
ORIGIN_SZ490   3.6851496  0.1288719    28.595  < 2e-16 ***
ORIGIN_SZ491   4.1880606  0.1285170    32.588  < 2e-16 ***
ORIGIN_SZ493  -1.1749485  0.2122468    -5.536 3.10e-08 ***
ORIGIN_SZ494   1.2353383  0.1428587     8.647  < 2e-16 ***
ORIGIN_SZ509   0.7461932  0.1334786     5.590 2.27e-08 ***
ORIGIN_SZ510   1.5711689  0.1308049    12.012  < 2e-16 ***
ORIGIN_SZ511   3.1384338  0.1287544    24.375  < 2e-16 ***
ORIGIN_SZ512   5.8131536  0.1283280    45.299  < 2e-16 ***
ORIGIN_SZ513   2.3201534  0.1306174    17.763  < 2e-16 ***
ORIGIN_SZ514   2.1168213  0.1340983    15.786  < 2e-16 ***
ORIGIN_SZ515   0.3959269  0.1995066     1.985 0.047197 *  
ORIGIN_SZ530  -0.1051997  0.1551347    -0.678 0.497697    
ORIGIN_SZ531   2.1354352  0.1298169    16.450  < 2e-16 ***
ORIGIN_SZ532   1.5154353  0.1307571    11.590  < 2e-16 ***
ORIGIN_SZ533   5.5870356  0.1283265    43.538  < 2e-16 ***
ORIGIN_SZ534   6.1313157  0.1283403    47.774  < 2e-16 ***
ORIGIN_SZ536   2.8040298  0.1308424    21.431  < 2e-16 ***
ORIGIN_SZ537   0.9009629  0.1773825     5.079 3.79e-07 ***
ORIGIN_SZ538   0.7319978  0.2448586     2.989 0.002795 ** 
ORIGIN_SZ539   0.1889885  1.0087880     0.187 0.851392    
ORIGIN_SZ551  -0.0802166  0.1389218    -0.577 0.563654    
ORIGIN_SZ552   2.3580939  0.1334290    17.673  < 2e-16 ***
ORIGIN_SZ553   1.9548006  0.1294728    15.098  < 2e-16 ***
ORIGIN_SZ554   5.4436879  0.1283357    42.418  < 2e-16 ***
ORIGIN_SZ555   4.7990556  0.1284894    37.350  < 2e-16 ***
ORIGIN_SZ559   1.3434072  0.1615223     8.317  < 2e-16 ***
ORIGIN_SZ560   3.7651027  0.1360404    27.676  < 2e-16 ***
ORIGIN_SZ561  -1.3562021  1.0094990    -1.343 0.179129    
ORIGIN_SZ572  -1.8802721  0.4282239    -4.391 1.13e-05 ***
ORIGIN_SZ573   0.4595648  0.1350553     3.403 0.000667 ***
ORIGIN_SZ574   1.8237630  0.1334734    13.664  < 2e-16 ***
ORIGIN_SZ575   6.5891046  0.1283083    51.354  < 2e-16 ***
ORIGIN_SZ576   5.0716420  0.1283839    39.504  < 2e-16 ***
ORIGIN_SZ578   0.3477660  0.1612539     2.157 0.031034 *  
ORIGIN_SZ582   0.7041518  0.1661964     4.237 2.27e-05 ***
ORIGIN_SZ583   0.6990335  0.4279531     1.633 0.102378    
ORIGIN_SZ584   4.2195182  0.1384317    30.481  < 2e-16 ***
ORIGIN_SZ593   1.5726840  0.1370901    11.472  < 2e-16 ***
ORIGIN_SZ594   1.5817148  0.1308139    12.091  < 2e-16 ***
ORIGIN_SZ595   1.8243274  0.1296637    14.070  < 2e-16 ***
ORIGIN_SZ596   4.8396880  0.1283597    37.704  < 2e-16 ***
ORIGIN_SZ597   2.0869369  0.1405580    14.848  < 2e-16 ***
ORIGIN_SZ603   1.3234042  0.1692391     7.820 5.29e-15 ***
ORIGIN_SZ604   1.7550673  0.2231652     7.864 3.71e-15 ***
ORIGIN_SZ615   0.2206062  0.1363140     1.618 0.105583    
ORIGIN_SZ616   1.2829325  0.1320936     9.712  < 2e-16 ***
ORIGIN_SZ617   2.1648199  0.1295708    16.708  < 2e-16 ***
ORIGIN_SZ618   5.5155910  0.1283468    42.974  < 2e-16 ***
ORIGIN_SZ620   2.6761751  0.1317617    20.311  < 2e-16 ***
ORIGIN_SZ637   1.0365392  0.1348394     7.687 1.50e-14 ***
ORIGIN_SZ638   4.9647838  0.1283593    38.679  < 2e-16 ***
ORIGIN_SZ657   2.8464046  0.1293749    22.001  < 2e-16 ***
ORIGIN_SZ658   4.3118367  0.1284518    33.568  < 2e-16 ***
ORIGIN_SZ659   4.5165384  0.1284061    35.174  < 2e-16 ***
ORIGIN_SZ660   5.6795444  0.1283370    44.255  < 2e-16 ***
ORIGIN_SZ662   5.6490277  0.1285159    43.956  < 2e-16 ***
ORIGIN_SZ677   2.7261064  0.1302955    20.922  < 2e-16 ***
ORIGIN_SZ678   0.5485599  0.1364948     4.019 5.85e-05 ***
ORIGIN_SZ679   5.5476455  0.1283465    43.224  < 2e-16 ***
ORIGIN_SZ680   5.7814553  0.1283449    45.046  < 2e-16 ***
ORIGIN_SZ681   4.4894509  0.1285630    34.920  < 2e-16 ***
ORIGIN_SZ699   2.2405240  0.1302647    17.200  < 2e-16 ***
ORIGIN_SZ700   4.4925825  0.1284369    34.979  < 2e-16 ***
ORIGIN_SZ701   4.4095803  0.1284380    34.332  < 2e-16 ***
ORIGIN_SZ702   5.4786781  0.1283432    42.688  < 2e-16 ***
ORIGIN_SZ704   2.5640128  0.1322083    19.394  < 2e-16 ***
ORIGIN_SZ722   2.2596147  0.1295939    17.436  < 2e-16 ***
ORIGIN_SZ725   2.0324175  0.1345148    15.109  < 2e-16 ***
ORIGIN_SZ741   2.7762245  0.1293588    21.461  < 2e-16 ***
ORIGIN_SZ743   2.9902364  0.1288881    23.200  < 2e-16 ***
ORIGIN_SZ744   5.0780767  0.1283722    39.557  < 2e-16 ***
ORIGIN_SZ761   2.3192045  0.1312756    17.667  < 2e-16 ***
ORIGIN_SZ762   4.2846462  0.1285059    33.342  < 2e-16 ***
ORIGIN_SZ763   1.5206569  0.1310794    11.601  < 2e-16 ***
ORIGIN_SZ764   5.5790146  0.1283416    43.470  < 2e-16 ***
ORIGIN_SZ765   4.7777270  0.1285630    37.163  < 2e-16 ***
ORIGIN_SZ767   6.6070303  0.1283571    51.474  < 2e-16 ***
ORIGIN_SZ772   1.3281976  0.1484396     8.948  < 2e-16 ***
ORIGIN_SZ784   1.3670047  0.1310123    10.434  < 2e-16 ***
ORIGIN_SZ785   4.6756446  0.1283700    36.423  < 2e-16 ***
ORIGIN_SZ786   4.3556867  0.1284389    33.913  < 2e-16 ***
ORIGIN_SZ787   4.9407778  0.1285040    38.448  < 2e-16 ***
ORIGIN_SZ788   5.7989893  0.1284141    45.159  < 2e-16 ***
ORIGIN_SZ789   4.2570120  0.1287218    33.071  < 2e-16 ***
ORIGIN_SZ803   0.5015402  0.1388696     3.612 0.000304 ***
ORIGIN_SZ804   5.3266445  0.1283676    41.495  < 2e-16 ***
ORIGIN_SZ805   5.2948963  0.1283446    41.255  < 2e-16 ***
ORIGIN_SZ806   5.0165155  0.1283678    39.079  < 2e-16 ***
ORIGIN_SZ807   6.0112348  0.1283728    46.826  < 2e-16 ***
ORIGIN_SZ808   4.2397833  0.1287592    32.928  < 2e-16 ***
ORIGIN_SZ809   6.0018355  0.1283491    46.762  < 2e-16 ***
ORIGIN_SZ810   5.2248926  0.1284356    40.681  < 2e-16 ***
ORIGIN_SZ814   3.1415033  0.1311682    23.950  < 2e-16 ***
ORIGIN_SZ819   5.2782677  0.1295641    40.739  < 2e-16 ***
ORIGIN_SZ824   1.4838427  0.1338729    11.084  < 2e-16 ***
ORIGIN_SZ826   2.9697079  0.1286813    23.078  < 2e-16 ***
ORIGIN_SZ827   4.9601290  0.1283584    38.643  < 2e-16 ***
ORIGIN_SZ828   5.3539759  0.1283522    41.713  < 2e-16 ***
ORIGIN_SZ829   5.4057517  0.1283801    42.107  < 2e-16 ***
ORIGIN_SZ830   5.6207518  0.1284042    43.774  < 2e-16 ***
ORIGIN_SZ831   6.5320399  0.1283357    50.898  < 2e-16 ***
ORIGIN_SZ832   5.6900053  0.1284630    44.293  < 2e-16 ***
ORIGIN_SZ835   3.0146096  0.1315282    22.920  < 2e-16 ***
ORIGIN_SZ844   0.4681830  0.1482345     3.158 0.001586 ** 
ORIGIN_SZ846   4.3998432  0.1284074    34.265  < 2e-16 ***
ORIGIN_SZ847   3.5040945  0.1286150    27.245  < 2e-16 ***
ORIGIN_SZ848   4.7250679  0.1283870    36.803  < 2e-16 ***
ORIGIN_SZ849   3.9550633  0.1285238    30.773  < 2e-16 ***
ORIGIN_SZ850   4.8690494  0.1284142    37.917  < 2e-16 ***
ORIGIN_SZ851   5.4253340  0.1283676    42.264  < 2e-16 ***
ORIGIN_SZ852   4.9548384  0.1284522    38.573  < 2e-16 ***
ORIGIN_SZ853   6.9365719  0.1283778    54.032  < 2e-16 ***
ORIGIN_SZ854   3.2237110  0.1308010    24.646  < 2e-16 ***
ORIGIN_SZ855   1.8416130  0.1353800    13.603  < 2e-16 ***
ORIGIN_SZ856   3.4683981  0.1296780    26.746  < 2e-16 ***
ORIGIN_SZ866   2.3172711  0.1310822    17.678  < 2e-16 ***
ORIGIN_SZ867   3.1976249  0.1288244    24.822  < 2e-16 ***
ORIGIN_SZ868   2.4882976  0.1294932    19.216  < 2e-16 ***
ORIGIN_SZ869   5.0233449  0.1284450    39.109  < 2e-16 ***
ORIGIN_SZ870   5.9104196  0.1283334    46.055  < 2e-16 ***
ORIGIN_SZ871   5.7459917  0.1284312    44.740  < 2e-16 ***
ORIGIN_SZ872   3.6190123  0.1287687    28.105  < 2e-16 ***
ORIGIN_SZ873   3.8506553  0.1286792    29.924  < 2e-16 ***
ORIGIN_SZ874   4.1234873  0.1285464    32.078  < 2e-16 ***
ORIGIN_SZ875   1.0949378  0.1438786     7.610 2.74e-14 ***
ORIGIN_SZ876   2.2269604  0.1318447    16.891  < 2e-16 ***
ORIGIN_SZ877   1.7867469  0.1341387    13.320  < 2e-16 ***
ORIGIN_SZ887   2.6306026  0.1289823    20.395  < 2e-16 ***
ORIGIN_SZ888   4.1253397  0.1284797    32.109  < 2e-16 ***
ORIGIN_SZ889   1.0535960  0.1320251     7.980 1.46e-15 ***
ORIGIN_SZ890   5.3284587  0.1283518    41.514  < 2e-16 ***
ORIGIN_SZ891   4.0268547  0.1287780    31.270  < 2e-16 ***
ORIGIN_SZ893   5.4681295  0.1283613    42.600  < 2e-16 ***
ORIGIN_SZ894   3.6741126  0.1287171    28.544  < 2e-16 ***
ORIGIN_SZ895   3.1918333  0.1291808    24.708  < 2e-16 ***
ORIGIN_SZ896   1.9784262  0.1315361    15.041  < 2e-16 ***
ORIGIN_SZ897   0.3264987  0.1420392     2.299 0.021525 *  
ORIGIN_SZ898   2.5017205  0.1320423    18.946  < 2e-16 ***
ORIGIN_SZ908   4.1028770  0.1288535    31.841  < 2e-16 ***
ORIGIN_SZ909   4.1157508  0.1284197    32.049  < 2e-16 ***
ORIGIN_SZ910   2.1546228  0.1293238    16.661  < 2e-16 ***
ORIGIN_SZ911   3.5439302  0.1285925    27.559  < 2e-16 ***
ORIGIN_SZ912   5.0794681  0.1283673    39.570  < 2e-16 ***
ORIGIN_SZ915   5.3722279  0.1283714    41.849  < 2e-16 ***
ORIGIN_SZ917   4.8669273  0.1285088    37.872  < 2e-16 ***
ORIGIN_SZ918   1.1976582  0.1338619     8.947  < 2e-16 ***
ORIGIN_SZ919   4.1125271  0.1286438    31.968  < 2e-16 ***
ORIGIN_SZ928   3.0455756  0.1289702    23.615  < 2e-16 ***
ORIGIN_SZ929   4.4978222  0.1283770    35.036  < 2e-16 ***
ORIGIN_SZ930   5.3657820  0.1283355    41.811  < 2e-16 ***
ORIGIN_SZ931   2.3162035  0.1293690    17.904  < 2e-16 ***
ORIGIN_SZ932   2.7694625  0.1294770    21.390  < 2e-16 ***
ORIGIN_SZ933   4.9357933  0.1284184    38.435  < 2e-16 ***
ORIGIN_SZ934   2.3306921  0.1296816    17.972  < 2e-16 ***
ORIGIN_SZ935   6.2496853  0.1283348    48.698  < 2e-16 ***
ORIGIN_SZ938  -1.6147369  0.2810424    -5.746 9.16e-09 ***
ORIGIN_SZ939   6.2324982  0.1283674    48.552  < 2e-16 ***
ORIGIN_SZ940   0.4374257  0.1465384     2.985 0.002835 ** 
ORIGIN_SZ949   2.7913292  0.1289455    21.647  < 2e-16 ***
ORIGIN_SZ950   4.9625752  0.1283901    38.652  < 2e-16 ***
ORIGIN_SZ951   5.8124044  0.1283280    45.293  < 2e-16 ***
ORIGIN_SZ952   2.5331222  0.1298464    19.509  < 2e-16 ***
ORIGIN_SZ953   3.7049359  0.1286368    28.802  < 2e-16 ***
ORIGIN_SZ954   2.5590881  0.1293518    19.784  < 2e-16 ***
ORIGIN_SZ955   4.8206888  0.1284189    37.539  < 2e-16 ***
ORIGIN_SZ956   3.2253869  0.1287870    25.044  < 2e-16 ***
ORIGIN_SZ957   5.3750668  0.1284242    41.854  < 2e-16 ***
ORIGIN_SZ959   1.4277296  0.1368777    10.431  < 2e-16 ***
ORIGIN_SZ961   1.4545836  0.1317928    11.037  < 2e-16 ***
ORIGIN_SZ962   6.7795777  0.1284728    52.771  < 2e-16 ***
ORIGIN_SZ970   3.3200875  0.1285368    25.830  < 2e-16 ***
ORIGIN_SZ971   4.6872948  0.1283546    36.518  < 2e-16 ***
ORIGIN_SZ972   4.5754058  0.1283801    35.640  < 2e-16 ***
ORIGIN_SZ974   4.2151790  0.1284972    32.804  < 2e-16 ***
ORIGIN_SZ975   3.8529613  0.1285929    29.962  < 2e-16 ***
ORIGIN_SZ976   2.5736286  0.1293162    19.902  < 2e-16 ***
ORIGIN_SZ977   5.7393499  0.1283528    44.715  < 2e-16 ***
ORIGIN_SZ978   5.6271784  0.1284820    43.797  < 2e-16 ***
ORIGIN_SZ982   2.5287621  0.1291557    19.579  < 2e-16 ***
ORIGIN_SZ983   7.4453499  0.1283646    58.002  < 2e-16 ***
ORIGIN_SZ984   7.4937210  0.1283984    58.363  < 2e-16 ***
ORIGIN_SZ991   3.3688268  0.1285932    26.198  < 2e-16 ***
ORIGIN_SZ992   3.6566784  0.1284744    28.462  < 2e-16 ***
ORIGIN_SZ993   3.4070732  0.1285045    26.513  < 2e-16 ***
ORIGIN_SZ994   2.9499187  0.1287275    22.916  < 2e-16 ***
ORIGIN_SZ995   4.3993511  0.1284162    34.259  < 2e-16 ***
ORIGIN_SZ996   4.1139306  0.1285169    32.011  < 2e-16 ***
ORIGIN_SZ997   2.8308973  0.1311854    21.579  < 2e-16 ***
ORIGIN_SZ998   5.8555628  0.1283574    45.619  < 2e-16 ***
ORIGIN_SZ999   5.8385828  0.1284204    45.465  < 2e-16 ***
ORIGIN_SZ1001  2.6713211  0.1306592    20.445  < 2e-16 ***
ORIGIN_SZ1003  5.3580592  0.1283740    41.738  < 2e-16 ***
ORIGIN_SZ1004  6.0074265  0.1283503    46.805  < 2e-16 ***
ORIGIN_SZ1011  1.3614625  0.1319247    10.320  < 2e-16 ***
ORIGIN_SZ1012  1.3964150  0.1321703    10.565  < 2e-16 ***
ORIGIN_SZ1013  2.1051864  0.1293209    16.279  < 2e-16 ***
ORIGIN_SZ1014  2.9620901  0.1287294    23.010  < 2e-16 ***
ORIGIN_SZ1015  2.4138370  0.1291886    18.685  < 2e-16 ***
ORIGIN_SZ1016  4.1555792  0.1284861    32.343  < 2e-16 ***
ORIGIN_SZ1018  4.7337897  0.1286573    36.794  < 2e-16 ***
ORIGIN_SZ1019  6.3123786  0.1283874    49.167  < 2e-16 ***
ORIGIN_SZ1023  4.3697894  0.1286306    33.972  < 2e-16 ***
ORIGIN_SZ1024  5.3946861  0.1283850    42.020  < 2e-16 ***
ORIGIN_SZ1025  0.5797077  0.1378446     4.206 2.60e-05 ***
ORIGIN_SZ1033  2.7628241  0.1287715    21.455  < 2e-16 ***
ORIGIN_SZ1034  3.7076914  0.1284935    28.855  < 2e-16 ***
ORIGIN_SZ1035  4.4197167  0.1284092    34.419  < 2e-16 ***
ORIGIN_SZ1036  2.4413074  0.1291227    18.907  < 2e-16 ***
ORIGIN_SZ1037  3.6810703  0.1284912    28.648  < 2e-16 ***
ORIGIN_SZ1043  2.7041178  0.1306722    20.694  < 2e-16 ***
ORIGIN_SZ1045  5.0036359  0.1283827    38.974  < 2e-16 ***
ORIGIN_SZ1046  4.8600249  0.1284019    37.850  < 2e-16 ***
ORIGIN_SZ1053  4.2805812  0.1284235    33.332  < 2e-16 ***
ORIGIN_SZ1054  3.1721677  0.1285584    24.675  < 2e-16 ***
ORIGIN_SZ1055  4.3164605  0.1284015    33.617  < 2e-16 ***
ORIGIN_SZ1056  3.3449664  0.1286159    26.007  < 2e-16 ***
ORIGIN_SZ1064  0.3641541  0.2684429     1.357 0.174927    
ORIGIN_SZ1066  4.7807553  0.1284028    37.232  < 2e-16 ***
ORIGIN_SZ1067  5.5362003  0.1284055    43.115  < 2e-16 ***
ORIGIN_SZ1074  3.6129635  0.1285148    28.113  < 2e-16 ***
ORIGIN_SZ1075  2.8313889  0.1287176    21.997  < 2e-16 ***
ORIGIN_SZ1076  3.0018657  0.1285817    23.346  < 2e-16 ***
ORIGIN_SZ1077  3.2299883  0.1286726    25.102  < 2e-16 ***
ORIGIN_SZ1079  4.6158888  0.1284070    35.947  < 2e-16 ***
ORIGIN_SZ1085 -1.6741102  0.2142978    -7.812 5.63e-15 ***
ORIGIN_SZ1087  4.5588996  0.1284355    35.496  < 2e-16 ***
ORIGIN_SZ1088  3.0334034  0.1288400    23.544  < 2e-16 ***
ORIGIN_SZ1094  1.6330476  0.1355237    12.050  < 2e-16 ***
ORIGIN_SZ1095  1.0000862  0.1337777     7.476 7.68e-14 ***
ORIGIN_SZ1096  2.9821904  0.1298896    22.959  < 2e-16 ***
ORIGIN_SZ1097  5.0906615  0.1283501    39.662  < 2e-16 ***
ORIGIN_SZ1098  2.6108029  0.1295066    20.160  < 2e-16 ***
ORIGIN_SZ1099  4.0158899  0.1284725    31.259  < 2e-16 ***
ORIGIN_SZ1105  1.7724246  0.1429485    12.399  < 2e-16 ***
ORIGIN_SZ1106  0.0153603  0.1613298     0.095 0.924147    
ORIGIN_SZ1107  2.3455815  0.1314244    17.847  < 2e-16 ***
ORIGIN_SZ1108  6.6047138  0.1283347    51.465  < 2e-16 ***
ORIGIN_SZ1109  3.9159747  0.1285455    30.464  < 2e-16 ***
ORIGIN_SZ1116  3.3236220  0.1285814    25.848  < 2e-16 ***
ORIGIN_SZ1117  1.7078013  0.1298638    13.151  < 2e-16 ***
ORIGIN_SZ1118  2.2710221  0.1295121    17.535  < 2e-16 ***
ORIGIN_SZ1119  3.2002136  0.1286806    24.869  < 2e-16 ***
ORIGIN_SZ1120  3.1584771  0.1289782    24.488  < 2e-16 ***
ORIGIN_SZ1129  5.0437158  0.1283974    39.282  < 2e-16 ***
ORIGIN_SZ1130  5.0856121  0.1283706    39.617  < 2e-16 ***
ORIGIN_SZ1131  4.4503068  0.1285519    34.619  < 2e-16 ***
ORIGIN_SZ1136  2.3574337  0.1290568    18.267  < 2e-16 ***
ORIGIN_SZ1138  0.9244841  0.1329645     6.953 3.58e-12 ***
ORIGIN_SZ1139  4.1524155  0.1284025    32.339  < 2e-16 ***
ORIGIN_SZ1141  4.1071977  0.1284565    31.973  < 2e-16 ***
ORIGIN_SZ1148  1.4188214  0.1384438    10.248  < 2e-16 ***
ORIGIN_SZ1149  2.9120967  0.1295611    22.477  < 2e-16 ***
ORIGIN_SZ1150  5.3044536  0.1283620    41.324  < 2e-16 ***
ORIGIN_SZ1151  4.2082018  0.1284711    32.756  < 2e-16 ***
ORIGIN_SZ1158  2.4313623  0.1287944    18.878  < 2e-16 ***
ORIGIN_SZ1159  3.9526476  0.1284212    30.779  < 2e-16 ***
ORIGIN_SZ1160  5.0104849  0.1283551    39.036  < 2e-16 ***
ORIGIN_SZ1171  5.8886025  0.1283717    45.871  < 2e-16 ***
ORIGIN_SZ1172  5.7411405  0.1283679    44.724  < 2e-16 ***
ORIGIN_SZ1173  2.8533942  0.1289138    22.134  < 2e-16 ***
ORIGIN_SZ1174 -1.6985984  0.2810566    -6.044 1.51e-09 ***
ORIGIN_SZ1178  2.7134960  0.1287189    21.081  < 2e-16 ***
ORIGIN_SZ1179  3.4454647  0.1284786    26.817  < 2e-16 ***
ORIGIN_SZ1180  4.5102659  0.1283724    35.134  < 2e-16 ***
ORIGIN_SZ1181  3.7584689  0.1284648    29.257  < 2e-16 ***
ORIGIN_SZ1183  2.8685027  0.1289376    22.247  < 2e-16 ***
ORIGIN_SZ1190  1.7003948  0.1368713    12.423  < 2e-16 ***
ORIGIN_SZ1192  4.6055665  0.1284622    35.852  < 2e-16 ***
ORIGIN_SZ1193  4.5964427  0.1284207    35.792  < 2e-16 ***
ORIGIN_SZ1194  2.8688838  0.1290323    22.234  < 2e-16 ***
ORIGIN_SZ1200  3.4689356  0.1284773    27.000  < 2e-16 ***
ORIGIN_SZ1201  3.3471765  0.1284958    26.049  < 2e-16 ***
ORIGIN_SZ1203  4.2403472  0.1284332    33.016  < 2e-16 ***
ORIGIN_SZ1204  3.9834418  0.1284700    31.007  < 2e-16 ***
ORIGIN_SZ1211  0.8732171  0.1653877     5.280 1.29e-07 ***
ORIGIN_SZ1214  5.3665337  0.1283778    41.803  < 2e-16 ***
ORIGIN_SZ1215  1.8296668  0.1344457    13.609  < 2e-16 ***
ORIGIN_SZ1216  0.9200164  0.1344061     6.845 7.64e-12 ***
ORIGIN_SZ1220  4.5147297  0.1283703    35.170  < 2e-16 ***
ORIGIN_SZ1221  4.1647047  0.1283842    32.439  < 2e-16 ***
ORIGIN_SZ1222  3.7853159  0.1286620    29.421  < 2e-16 ***
ORIGIN_SZ1223  2.1485159  0.1293868    16.605  < 2e-16 ***
ORIGIN_SZ1224  3.0637462  0.1287171    23.802  < 2e-16 ***
ORIGIN_SZ1231 -0.5785710  0.1931693    -2.995 0.002743 ** 
ORIGIN_SZ1232  1.6020338  0.1577109    10.158  < 2e-16 ***
ORIGIN_SZ1235  2.8891562  0.1290306    22.391  < 2e-16 ***
ORIGIN_SZ1236  2.6706813  0.1291650    20.677  < 2e-16 ***
ORIGIN_SZ1241  2.6749712  0.1287175    20.782  < 2e-16 ***
ORIGIN_SZ1242  3.3558722  0.1284773    26.120  < 2e-16 ***
ORIGIN_SZ1243  4.0336892  0.1283948    31.416  < 2e-16 ***
ORIGIN_SZ1246  3.2552150  0.1286250    25.308  < 2e-16 ***
ORIGIN_SZ1256  4.2865697  0.1285178    33.354  < 2e-16 ***
ORIGIN_SZ1257  5.1521683  0.1284247    40.118  < 2e-16 ***
ORIGIN_SZ1258  3.1495727  0.1290847    24.399  < 2e-16 ***
ORIGIN_SZ1262  3.0838272  0.1285660    23.986  < 2e-16 ***
ORIGIN_SZ1263  3.6926366  0.1284114    28.756  < 2e-16 ***
ORIGIN_SZ1264  2.6701258  0.1288620    20.721  < 2e-16 ***
ORIGIN_SZ1265  2.4338921  0.1291175    18.850  < 2e-16 ***
ORIGIN_SZ1266  3.8381859  0.1285277    29.863  < 2e-16 ***
ORIGIN_SZ1267  1.9652566  0.1309093    15.012  < 2e-16 ***
ORIGIN_SZ1272 -0.6005707  0.1784891    -3.365 0.000766 ***
ORIGIN_SZ1273  3.7329503  0.1288496    28.971  < 2e-16 ***
ORIGIN_SZ1277  5.3300196  0.1283783    41.518  < 2e-16 ***
ORIGIN_SZ1278  2.7968417  0.1289899    21.683  < 2e-16 ***
ORIGIN_SZ1283  4.9383464  0.1283620    38.472  < 2e-16 ***
ORIGIN_SZ1284  4.0964869  0.1283938    31.906  < 2e-16 ***
ORIGIN_SZ1285  4.8056051  0.1283510    37.441  < 2e-16 ***
ORIGIN_SZ1286  3.0869296  0.1286914    23.987  < 2e-16 ***
ORIGIN_SZ1289  1.3410831  0.1362765     9.841  < 2e-16 ***
ORIGIN_SZ1293  0.3168305  0.1480128     2.141 0.032309 *  
ORIGIN_SZ1294  5.1414463  0.1284542    40.026  < 2e-16 ***
ORIGIN_SZ1295  2.6145693  0.1299450    20.121  < 2e-16 ***
ORIGIN_SZ1298  4.3456426  0.1284881    33.821  < 2e-16 ***
ORIGIN_SZ1299  4.5959685  0.1284570    35.778  < 2e-16 ***
ORIGIN_SZ1304  4.7072140  0.1283653    36.670  < 2e-16 ***
ORIGIN_SZ1305  4.1335815  0.1283719    32.200  < 2e-16 ***
ORIGIN_SZ1307  0.7500065  0.1331626     5.632 1.78e-08 ***
ORIGIN_SZ1308  3.9720413  0.1284550    30.922  < 2e-16 ***
ORIGIN_SZ1310 -0.7703709  0.2141972    -3.597 0.000322 ***
ORIGIN_SZ1316  1.5467810  0.1329477    11.635  < 2e-16 ***
ORIGIN_SZ1317  3.3278269  0.1288029    25.837  < 2e-16 ***
ORIGIN_SZ1318  2.6193558  0.1292355    20.268  < 2e-16 ***
ORIGIN_SZ1319  5.5800661  0.1283719    43.468  < 2e-16 ***
ORIGIN_SZ1320  4.0232570  0.1286368    31.276  < 2e-16 ***
ORIGIN_SZ1325  2.3380308  0.1288702    18.143  < 2e-16 ***
ORIGIN_SZ1326  4.3044463  0.1283701    33.532  < 2e-16 ***
ORIGIN_SZ1327  4.1268654  0.1283838    32.145  < 2e-16 ***
ORIGIN_SZ1328  2.7620098  0.1286476    21.470  < 2e-16 ***
ORIGIN_SZ1329  3.0506402  0.1288063    23.684  < 2e-16 ***
ORIGIN_SZ1330  3.8621532  0.1285638    30.041  < 2e-16 ***
ORIGIN_SZ1331 -1.2440279  0.2810167    -4.427 9.56e-06 ***
ORIGIN_SZ1333  3.2282984  0.1288720    25.050  < 2e-16 ***
ORIGIN_SZ1334  3.5087803  0.1287888    27.244  < 2e-16 ***
ORIGIN_SZ1335  3.0465284  0.1293834    23.547  < 2e-16 ***
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ORIGIN_SZ1390  3.3067639  0.1284756    25.738  < 2e-16 ***
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ORIGIN_SZ1417  3.7745313  0.1284677    29.381  < 2e-16 ***
ORIGIN_SZ1418  4.5713200  0.1283965    35.603  < 2e-16 ***
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ORIGIN_SZ1436  1.3014336  0.1303414     9.985  < 2e-16 ***
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ORIGIN_SZ1438  5.0522959  0.1283502    39.363  < 2e-16 ***
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ORIGIN_SZ1440  2.1137319  0.1298160    16.283  < 2e-16 ***
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ORIGIN_SZ1444  5.0405899  0.1285228    39.219  < 2e-16 ***
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ORIGIN_SZ1453  3.0191038  0.1285082    23.493  < 2e-16 ***
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ORIGIN_SZ1455  3.3886706  0.1285616    26.358  < 2e-16 ***
ORIGIN_SZ1456  3.8310384  0.1284343    29.829  < 2e-16 ***
ORIGIN_SZ1457  4.6346391  0.1283695    36.104  < 2e-16 ***
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ORIGIN_SZ1459  4.4130417  0.1283861    34.373  < 2e-16 ***
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ORIGIN_SZ1476  3.8272335  0.1284468    29.796  < 2e-16 ***
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ORIGIN_SZ1480  5.6525484  0.1283348    44.045  < 2e-16 ***
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ORIGIN_SZ1482  3.2109390  0.1289469    24.901  < 2e-16 ***
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ORIGIN_SZ1495  2.8544767  0.1285226    22.210  < 2e-16 ***
ORIGIN_SZ1496  3.9820085  0.1283819    31.017  < 2e-16 ***
ORIGIN_SZ1497  4.2639611  0.1284014    33.208  < 2e-16 ***
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ORIGIN_SZ1502  4.4058222  0.1283888    34.316  < 2e-16 ***
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ORIGIN_SZ1517  3.6509048  0.1284673    28.419  < 2e-16 ***
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ORIGIN_SZ1537  1.8835853  0.1293558    14.561  < 2e-16 ***
ORIGIN_SZ1538  3.4387325  0.1284423    26.773  < 2e-16 ***
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ORIGIN_SZ1559  4.1062899  0.1283844    31.984  < 2e-16 ***
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ORIGIN_SZ1793  2.0307331  0.1289694    15.746  < 2e-16 ***
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ORIGIN_SZ1818  5.1927141  0.1283798    40.448  < 2e-16 ***
ORIGIN_SZ1819  5.7273655  0.1283578    44.620  < 2e-16 ***
ORIGIN_SZ1820  2.8647670  0.1327721    21.577  < 2e-16 ***
ORIGIN_SZ1832  4.5074899  0.1284059    35.103  < 2e-16 ***
ORIGIN_SZ1833  3.9377954  0.1284330    30.660  < 2e-16 ***
ORIGIN_SZ1834  4.2644257  0.1283825    33.217  < 2e-16 ***
ORIGIN_SZ1835  3.5659499  0.1285070    27.749  < 2e-16 ***
ORIGIN_SZ1837  0.7690953  0.1419186     5.419 5.98e-08 ***
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ORIGIN_SZ1842  6.4694885  0.1287331    50.255  < 2e-16 ***
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ORIGIN_SZ1854  4.5606620  0.1283856    35.523  < 2e-16 ***
ORIGIN_SZ1855  4.8503696  0.1283699    37.784  < 2e-16 ***
ORIGIN_SZ1858  2.6470639  0.1307361    20.247  < 2e-16 ***
ORIGIN_SZ1860  5.9892312  0.1285733    46.582  < 2e-16 ***
ORIGIN_SZ1861  5.4449058  0.1283857    42.411  < 2e-16 ***
ORIGIN_SZ1874  4.5098276  0.1284380    35.113  < 2e-16 ***
ORIGIN_SZ1875  2.8330493  0.1289053    21.978  < 2e-16 ***
ORIGIN_SZ1876  4.9442722  0.1287017    38.417  < 2e-16 ***
ORIGIN_SZ1877  4.5626564  0.1283895    35.538  < 2e-16 ***
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ORIGIN_SZ1882  5.2992540  0.1283965    41.273  < 2e-16 ***
ORIGIN_SZ1883  5.8024625  0.1285970    45.121  < 2e-16 ***
ORIGIN_SZ1895  4.6351599  0.1283881    36.103  < 2e-16 ***
ORIGIN_SZ1896  2.9955843  0.1286416    23.286  < 2e-16 ***
ORIGIN_SZ1897  3.6604153  0.1285855    28.467  < 2e-16 ***
ORIGIN_SZ1898 -0.6599083  0.1778219    -3.711 0.000206 ***
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ORIGIN_SZ1903  5.1887292  0.1285909    40.351  < 2e-16 ***
ORIGIN_SZ1917  3.2192269  0.1286185    25.029  < 2e-16 ***
ORIGIN_SZ1918  4.6602616  0.1284272    36.287  < 2e-16 ***
ORIGIN_SZ1919  4.5380961  0.1284026    35.343  < 2e-16 ***
ORIGIN_SZ1922  2.3087644  0.1312319    17.593  < 2e-16 ***
ORIGIN_SZ1924  5.0214296  0.1286230    39.040  < 2e-16 ***
ORIGIN_SZ1937  4.0309239  0.1284751    31.375  < 2e-16 ***
ORIGIN_SZ1938  4.6755113  0.1283771    36.420  < 2e-16 ***
ORIGIN_SZ1939  5.2285909  0.1283681    40.731  < 2e-16 ***
ORIGIN_SZ1942  1.2796556  0.1345281     9.512  < 2e-16 ***
ORIGIN_SZ1959  3.6051332  0.1285704    28.040  < 2e-16 ***
ORIGIN_SZ1960  5.5903041  0.1283372    43.559  < 2e-16 ***
ORIGIN_SZ1961  3.9886969  0.1284784    31.046  < 2e-16 ***
ORIGIN_SZ1962  5.0143993  0.1283745    39.061  < 2e-16 ***
ORIGIN_SZ1964  0.2513609  0.1525944     1.647 0.099507 .  
ORIGIN_SZ1979  3.9080186  0.1285324    30.405  < 2e-16 ***
ORIGIN_SZ1980  2.1544333  0.1290272    16.698  < 2e-16 ***
ORIGIN_SZ1981  4.6365812  0.1283758    36.117  < 2e-16 ***
ORIGIN_SZ1982  3.4508601  0.1287496    26.803  < 2e-16 ***
ORIGIN_SZ1983  4.9242564  0.1283778    38.358  < 2e-16 ***
ORIGIN_SZ1984  3.6431087  0.1285296    28.345  < 2e-16 ***
ORIGIN_SZ1985  4.4345906  0.1284509    34.524  < 2e-16 ***
ORIGIN_SZ2001  4.2184167  0.1284302    32.846  < 2e-16 ***
ORIGIN_SZ2002  4.7630499  0.1283596    37.107  < 2e-16 ***
ORIGIN_SZ2003  4.3973471  0.1283983    34.248  < 2e-16 ***
ORIGIN_SZ2004  5.0199149  0.1283720    39.104  < 2e-16 ***
ORIGIN_SZ2005  4.8956048  0.1283712    38.136  < 2e-16 ***
ORIGIN_SZ2006  3.4575991  0.1286229    26.882  < 2e-16 ***
ORIGIN_SZ2007  2.8498303  0.1292845    22.043  < 2e-16 ***
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ORIGIN_SZ2023  5.0117105  0.1283577    39.045  < 2e-16 ***
ORIGIN_SZ2024  4.3242920  0.1283898    33.681  < 2e-16 ***
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ORIGIN_SZ2026  3.0208249  0.1287979    23.454  < 2e-16 ***
ORIGIN_SZ2027  5.3158698  0.1283666    41.412  < 2e-16 ***
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ORIGIN_SZ2044  4.6274993  0.1283729    36.047  < 2e-16 ***
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ORIGIN_SZ2046  4.5807504  0.1283630    35.686  < 2e-16 ***
ORIGIN_SZ2047  4.3432419  0.1284089    33.824  < 2e-16 ***
ORIGIN_SZ2048  4.6367218  0.1283862    36.115  < 2e-16 ***
ORIGIN_SZ2049  3.7174760  0.1289577    28.827  < 2e-16 ***
ORIGIN_SZ2064  4.5752841  0.1283961    35.634  < 2e-16 ***
ORIGIN_SZ2065  3.0666127  0.1285674    23.852  < 2e-16 ***
ORIGIN_SZ2066  2.4252096  0.1303253    18.609  < 2e-16 ***
ORIGIN_SZ2067  5.8329188  0.1283346    45.451  < 2e-16 ***
ORIGIN_SZ2068  4.1693543  0.1284817    32.451  < 2e-16 ***
ORIGIN_SZ2069  4.5189874  0.1284176    35.190  < 2e-16 ***
ORIGIN_SZ2085  3.4872368  0.1286312    27.110  < 2e-16 ***
ORIGIN_SZ2086  4.9165833  0.1283673    38.301  < 2e-16 ***
ORIGIN_SZ2087  3.5650820  0.1284901    27.746  < 2e-16 ***
ORIGIN_SZ2088  4.2418364  0.1283844    33.040  < 2e-16 ***
ORIGIN_SZ2089  3.9807971  0.1284591    30.989  < 2e-16 ***
ORIGIN_SZ2090  5.5293310  0.1283511    43.080  < 2e-16 ***
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ORIGIN_SZ2106  1.8173872  0.1296309    14.020  < 2e-16 ***
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ORIGIN_SZ2108  4.2092619  0.1284188    32.778  < 2e-16 ***
ORIGIN_SZ2109  4.2850041  0.1283829    33.377  < 2e-16 ***
ORIGIN_SZ2110  3.7216005  0.1285029    28.961  < 2e-16 ***
ORIGIN_SZ2111  1.1562654  0.1339491     8.632  < 2e-16 ***
ORIGIN_SZ2128  2.6600982  0.1290106    20.619  < 2e-16 ***
ORIGIN_SZ2129  2.0823288  0.1291362    16.125  < 2e-16 ***
ORIGIN_SZ2130  4.7533655  0.1283628    37.031  < 2e-16 ***
ORIGIN_SZ2131  5.0766995  0.1283819    39.544  < 2e-16 ***
ORIGIN_SZ2132  4.0403178  0.1284398    31.457  < 2e-16 ***
ORIGIN_SZ2148  3.5612816  0.1287767    27.655  < 2e-16 ***
ORIGIN_SZ2149  2.1561798  0.1291392    16.697  < 2e-16 ***
ORIGIN_SZ2150  4.6576297  0.1283840    36.279  < 2e-16 ***
ORIGIN_SZ2151  5.1132952  0.1283533    39.838  < 2e-16 ***
ORIGIN_SZ2152  5.3033728  0.1283595    41.317  < 2e-16 ***
ORIGIN_SZ2153  4.6356684  0.1284211    36.097  < 2e-16 ***
ORIGIN_SZ2171  3.6332254  0.1284957    28.275  < 2e-16 ***
ORIGIN_SZ2172  3.7323010  0.1285285    29.039  < 2e-16 ***
ORIGIN_SZ2173  4.2773750  0.1284030    33.312  < 2e-16 ***
ORIGIN_SZ2174  4.5619158  0.1284115    35.526  < 2e-16 ***
ORIGIN_SZ2191  3.1114873  0.1287109    24.174  < 2e-16 ***
ORIGIN_SZ2192  3.0217062  0.1287488    23.470  < 2e-16 ***
ORIGIN_SZ2193  3.9655386  0.1284480    30.873  < 2e-16 ***
ORIGIN_SZ2194  4.4091324  0.1284040    34.338  < 2e-16 ***
ORIGIN_SZ2195  3.7733700  0.1298799    29.053  < 2e-16 ***
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ORIGIN_SZ2236  2.8109067  0.1289231    21.803  < 2e-16 ***
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ORIGIN_SZ2256  0.5871896  0.1350322     4.349 1.37e-05 ***
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ORIGIN_SZ2258  2.1509525  0.1292432    16.643  < 2e-16 ***
ORIGIN_SZ2259  1.8622755  0.1325604    14.049  < 2e-16 ***
ORIGIN_SZ2277  2.6688346  0.1334693    19.996  < 2e-16 ***
ORIGIN_SZ2278  3.7780627  0.1288226    29.328  < 2e-16 ***
ORIGIN_SZ2279  1.7018214  0.1298144    13.110  < 2e-16 ***
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ORIGIN_SZ2297  4.6422258  0.1286067    36.096  < 2e-16 ***
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ORIGIN_SZ2318  2.6638788  0.1295632    20.560  < 2e-16 ***
ORIGIN_SZ2319  3.3334151  0.1287048    25.900  < 2e-16 ***
ORIGIN_SZ2322  3.3271113  0.1288204    25.828  < 2e-16 ***
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ORIGIN_SZ2343  2.5327363  0.1290309    19.629  < 2e-16 ***
ORIGIN_SZ2361  3.4319983  0.1288172    26.642  < 2e-16 ***
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ORIGIN_SZ2384  2.5353185  0.1291383    19.633  < 2e-16 ***
ORIGIN_SZ2405  2.2192627  0.1291656    17.182  < 2e-16 ***
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DESTIN_SZ68    0.0845370  0.0547088     1.545 0.122293    
DESTIN_SZ86    0.4635816  0.0610670     7.591 3.17e-14 ***
DESTIN_SZ87   -3.1422407  0.0988168   -31.799  < 2e-16 ***
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DESTIN_SZ154  -2.2477477  0.0577318   -38.934  < 2e-16 ***
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DESTIN_SZ406   0.2236529  0.0513142     4.358 1.31e-05 ***
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DESTIN_SZ408   1.1897568  0.0513201    23.183  < 2e-16 ***
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DESTIN_SZ536  -1.7144512  0.0540887   -31.697  < 2e-16 ***
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log(dist)     -1.6146098  0.0002885 -5595.605  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 101244825  on 64961  degrees of freedom
Residual deviance:  25322395  on 63326  degrees of freedom
AIC: 25684037

Number of Fisher Scoring iterations: 10

In the Destin Constrained Model, the coefficients for five variables are all NA (Not Available), meaning that suitable coefficient values could not be calculated during the model fitting process. The specific cause requires further examination

For the purpose of model comparison, we decided to exclude Destin Constrained Model. This ensures that, for a reliable comparison of model performance, the comparison is based on reliable estimates, avoiding the influence of variables with coefficients that could not be computed.

9 Model Comparison

First of all, let us create a list called model_list by using the code chunk below. It contains all our fitted models for all four variations of gravity model.

model_list <- list(
  Unconstrained = uncSIM,
  Origin_Constrained = orcSIM,
  Doubly_Constrained = dbcSIM)

Next, we will compute the RMSE of all the models in model_list file by using the code chunk below.

The RMSE values provide a measure of the goodness of fit for each model. A lower RMSE indicates better model performance in terms of how well the model predictions align with the observed values. In this comparison, the Doubly Constrained Model has the lowest RMSE (1219.291), suggesting that it performs better in terms of minimizing prediction errors compared to the Unconstrained and Origin Constrained models.

compare_performance(model_list,
                    metrics = "RMSE")
# Comparison of Model Performance Indices

Name               | Model |     RMSE
-------------------------------------
Unconstrained      |   glm | 1773.898
Origin_Constrained |   glm | 1653.729
Doubly_Constrained |   glm | 1219.291

9.1 Visualising the fitted values

We will do a plot to visualise the observed values and the fitted values.

Firstly we will extract the fitted values from each model by using the code chunk below.

df <- as.data.frame(uncSIM$fitted.values) %>%
  round(digits = 0)

Next, we will join the values to SIM_data data frame:

inter_zonal_flow <- inter_zonal_flow %>%
  cbind(df) %>%
  rename(uncTRIPS = "uncSIM$fitted.values")

Repeat the same step by for Origin Constrained SIM:

df <- as.data.frame(orcSIM$fitted.values) %>%
  round(digits = 0)
inter_zonal_flow <- inter_zonal_flow %>%
  cbind(df) %>%
  rename(orcTRIPS = "orcSIM$fitted.values")

Repeat the same step by for Doubly Constrained SIM:

df<- as.data.frame(dbcSIM$fitted.values) %>%   
  round(digits = 0)
inter_zonal_flow <- inter_zonal_flow %>%
  cbind(df) %>%
  rename(dbcTRIPS = "dbcSIM$fitted.values")

We will create scatter plots displaying the relationship between actual and predicted values through linear fitting and compare each other.

Code
unc_p <- ggplot(data = inter_zonal_flow,
                aes(x = uncTRIPS,
                    y = MORNING_PEAK)) +
  geom_point(color = "darkgreen") +
  geom_smooth(method = lm, color = "yellow") +
  theme_minimal()
  coord_cartesian(xlim=c(0,100000),
                  ylim=c(0,100000))
<ggproto object: Class CoordCartesian, Coord, gg>
    aspect: function
    backtransform_range: function
    clip: on
    default: FALSE
    distance: function
    expand: TRUE
    is_free: function
    is_linear: function
    labels: function
    limits: list
    modify_scales: function
    range: function
    render_axis_h: function
    render_axis_v: function
    render_bg: function
    render_fg: function
    setup_data: function
    setup_layout: function
    setup_panel_guides: function
    setup_panel_params: function
    setup_params: function
    train_panel_guides: function
    transform: function
    super:  <ggproto object: Class CoordCartesian, Coord, gg>
Code
orc_p <- ggplot(data = inter_zonal_flow,
                aes(x = orcTRIPS,
                    y = MORNING_PEAK)) +
  geom_point(color = "darkgreen") +
  geom_smooth(method = lm, color = "yellow") +
  theme_minimal()+
  coord_cartesian(xlim=c(0,100000),
                  ylim=c(0,100000))

dbc_p <- ggplot(data = inter_zonal_flow,
                aes(x = dbcTRIPS,
                    y = MORNING_PEAK)) +
  geom_point(color = "darkgreen") +
  geom_smooth(method = lm, color = "yellow") +
  theme_minimal()+
  coord_cartesian(xlim=c(0,100000),
                  ylim=c(0,100000))

ggarrange( unc_p, orc_p, dbc_p,
          ncol = 2,
          nrow = 2)

The results indicate that for the Doubly Constrained Model, the scatter plot points closely follow the fitted curve. This tight distribution suggests strong linear correlation between the actual values and the model predictions. The model appears to accurately capture the variation trends in the actual values, making it well-suited for describing underlying relationships.

10 Conclusion

In the comparison of three spatial interaction models (UNC, ORC, DBC), insights are provided through the examination of R-squared values. Specifically, the DBC model stands out with an R-squared value of 59%, signifying superior explanatory power among the considered models. Additionally, the comparison of Root Mean Square Error (RMSE) reveals that the DBC model has the lowest RMSE at 1219.291. This implies that the DBC model demonstrates the least average deviation in predictions compared to observed values, showcasing higher predictive accuracy.

According to the UNC and ORC models, the p-values of all the variables are very small (<0.05) , indicating that their contributions in the models are highly statistically significant. Considering both models, Entertainment consistently emerges as the variable with the most significant positive impact, followed by School and HDB. Distance consistently exhibits the most significant negative impact in both the UNC and ORC models.